%I M3664 N1490 #255 Nov 11 2024 10:33:05

%S 1,4,36,400,4900,63504,853776,11778624,165636900,2363904400,

%T 34134779536,497634306624,7312459672336,108172480360000,

%U 1609341595560000,24061445010950400,361297635242552100,5445717990022688400,82358080713306090000,1249287673091590440000

%N a(n) = binomial(2n, n)^2.

%C a(n) is the number of monotonic paths (only moving N and E) in the lattice [0..2n] X [0..2n] that contain the points (0,0), (n,n) and (2n,2n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002

%C This is the Taylor expansion of a special point on a curve described by Beauville. - _Matthijs Coster_, Apr 28 2004

%C Expansion of K(k) / (Pi/2) in powers of m/16 = (k/4)^2, where K(k) is the complete elliptic integral of the first kind evaluated at k. - _Michael Somos_, Mar 04 2003

%C Square lattice walks that start and end at origin after 2n steps. - _Gareth McCaughan_ and _Michael Somos_, Jun 12 2004

%C If A is a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic) then a(n)=E[(tr(A^k))^{2n}] for any k > 4. - _Andrew V. Sutherland_, Apr 01 2008

%C From _R. H. Hardin_, Feb 03 2016 and _R. J. Mathar_, Feb 18 2016: (Start)

%C Also, number of 2 X (2n) arrays of permutations of 2n copies of 0 or 1 with row sums equal.

%C For example, some solutions for n=3:

%C 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 1 0 0 0

%C 1 0 0 0 1 1 1 1 0 1 0 0 0 0 0 1 1 1 0 0 1 1 0 1

%C There is a simple combinatorial argument to show that this is a(n): We have 2n copies of 0's and 1's and need equal row sums. Therefore there must be n 1's in each of the two rows. Otherwise there are no constraints, so there are C(2n,n) ways of placing the 1's in the first row and independently C(2n,n) ways of placing the 1's in the second. The product is clearly C(2n,n)^2. (End)

%C Also the even part of the bisection of A241530. One half of the odd part is given in A000894. - _Wolfdieter Lang_, Sep 06 2016

%C From _Peter Bala_, Jan 26 2018: (Start)

%C Let S = {[1,0,0], [0,1,0], [1,0,1], [0,1,1]} be a set of four column vectors. Then a(n) equals the number of 3 X k arrays whose columns belong to the set S and whose row sums are all equal to n (apply Eger, Theorem 3). An example is given below. Equivalently, a(n) equals the number of lattice paths from (0,0,0) to (n,n,n) using steps (1,0,0), (0,1,0), (1,0,1) and (0,1,1).

%C The o.g.f. for the sequence equals the diagonal of the rational function 1/(1 - (x + y + x*z + y*z)).

%C Row sums of A069466. (End)

%C Also, the constant term in the expansion of (x + 1/x + y + 1/y)^(2n). - _Christopher J. Smyth_, Sep 26 2018

%C Number of ways to place 2n^2 nonattacking pawns on a 2n x 2n board. - _Tricia Muldoon Brown_, Dec 12 2018

%C For n>0, a(n) is the number of Littlewood polynomials of degree 4n-1 that have a closed Lill path. A polynomial p(x) has a closed Lill path if and only if p(x) is divisible by x^(2)+1. - _Raul Prisacariu_, Aug 28 2024

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 591,828.

%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.

%D Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

%D Leonard Lipshitz and A. van der Poorten. "Rational functions, diagonals, automata and arithmetic." In Number Theory, Richard A. Mollin, ed., Walter de Gruyter, Berlin (1990): 339-358.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002894/b002894.txt">Table of n, a(n) for n = 0..100</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H R. Bacher, <a href="https://www-fourier.ujf-grenoble.fr/sites/default/files/ref_478.pdf">Meander algebras</a>, Institut Fourier, 1999.

%H E. Barcucci, A. Frosini and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.disc.2005.01.006">On directed-convex polyominoes in a rectangle</a>, Discr. Math., 298 (2005). 62-78.

%H Arnaud Beauville, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k64348089/f201.image.r">Les familles stables de courbes elliptiques sur P^1 admettant quatre fibres singulières</a>, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982, 657-660. MR0664643 (83h:14008)

%H Alin Bostan, Armin Straub, and Sergey Yurkevich, <a href="https://arxiv.org/abs/2212.10116">On the representability of sequences as constant terms</a>, arXiv:2212.10116 [math.NT], 2022.

%H Tricia Muldoon Brown, <a href="https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i3p21/7881">The Problem of Pawns</a>, The Electronic Journal of Combinatorics (2019) Vol. 26, Issue 3, #P3.21. Also <a href="https://arxiv.org/abs/1811.09606">arXiv:1811.09606</a>, [math.CO], 2018.

%H John Maxwell Campbell, <a href="https://dx.doi.org/10.1216/RMJ-2019-49-8-2513">New series involving harmonic numbers and squared central binomial coefficients</a>, Rocky Mountain J. Math., 49 (2019), 2513-2544.

%H C. Domb, <a href="http://dx.doi.org/10.1080/00018736000101199">On the theory of cooperative phenomena in crystals</a>, Advances in Phys., 9 (1960), 149-361.

%H Steffen Eger, <a href="http://arxiv.org/abs/1511.00622">On the Number of Many-to-Many Alignments of N Sequences</a>, arXiv:1511.00622 [math.CO], 2015.

%H Murray Elder, <a href="http://carma.newcastle.edu.au/pdf/retreat2011/posters/MurrayElder-poster-A0.pdf">Cogrowth</a>, 2011.

%H M. Elder, A. Rechnitzer, E. J. Janse van Rensburg, and T. Wong, <a href="http://arxiv.org/abs/1309.4184">The cogrowth series for BS(N,N) is D-finite</a>, arXiv:1309.4184 [math.GR], 2013.

%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 90.

%H Florian Fürnsinn and Sergey Yurkevich, <a href="https://arxiv.org/abs/2308.12855">Algebraicity of hypergeometric functions with arbitrary parameters</a>, arXiv:2308.12855 [math.CA], 2023.

%H Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</a>, arXiv:0803.4462 [math.NT], 2010.

%H Markus Kuba and Alois Panholzer, <a href="https://arxiv.org/abs/2411.03930">Lattice paths and the diagonal of the cube</a>, arXiv:2411.03930 [math.CO], 2024. See p. 14.

%H L. Lipshitz and A. J. van der Poorten, <a href="http://citeseerx.ist.psu.edu/pdf/dfa9976c3141d5a40b1cf14231cbcbc85504b61e">Rational functions, diagonals, automata and arithmetic</a>

%H Raul Prisacariu, <a href="https://web.archive.org/web/20240803085130/https://www.raulprisacariu.com/math/littlewood-polynomials-of-degree-n-with-closed-lill-paths/">Littlewood Polynomials of Degree n with Closed Lill Paths</a>

%H Eric M. Rains, <a href="http://dx.doi.org/10.1007/s004400050084">High powers of random elements of compact Lie groups</a>, Probability Theory and Related Fields 107 (1997), 219-241.

%H Grzegorz Siudem and Agata Fronczak, <a href="https://arxiv.org/abs/2007.16132">Bell polynomials in the series expansions of the Ising model</a>, arXiv:2007.16132 [math-ph], 2020.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LatticePath.html">Lattice Path.</a>

%H D. Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/tex/AperylikeRecEqs/fulltext.pdf">Integral solutions of Apéry-like recurrence equations</a>. See line G in sporadic solutions table of page 5.

%F D-finite with recurrence: (n+1)^2*a(n+1) = 4*(2*n + 1)^2*a(n). - _Matthijs Coster_, Apr 28 2004

%F a(n) ~ Pi^(-1)*n^(-1)*2^(4*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002

%F G.f.: F(1/2, 1/2; 1; 16*x) = 1 / AGM(1, (1 - 16*x)^(1/2)) = K(4*sqrt(x)) / (Pi/2), where AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre. - _Michael Somos_, Mar 04 2003

%F G.f.: 2*EllipticK(4*sqrt(x))/Pi, using Maple's convention for elliptic integrals.

%F E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2x)^2.

%F a(n) = A000984(n)^2 = ((2*n)!/(n!)^2)^2 = (((2*n)!)^2)/((n!)^4). a(n) = A000984(n)^2 = ((((2^n)*(2*n-1)!!)/(n!)))^2 = (((2^(2*n))*(2*n-1)!!)^2)/(n!)^2). - _Jonathan Vos Post_, Jun 17 2007

%F E.g.f.: (BesselI(0, 2x))^2=1+(2*x^2)/(U(0)-2*x^2); U(k)=(2*x^2)*(2*k+1)+(k+1)^3-(2*x^2)*(2*k+3)*((k+1)^3)/U(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Nov 23 2011

%F In generally, for (BesselI(b, 2x))^2=((x^(2*b))/(GAMMA(b+1))^2)*(1+(2*x^2)*(2*b+1)/(Q(0)-(2*x^2)*(2*b+1)); Q(k)=(2*x^2)*(2*k+2*b+1)+(k+1)*(k+b+1)*(k+2*b+1)-(2*x^2)*(k+1)*(k+b+1)*(k+2*b+1)*(2*k+2*b+3)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Nov 23 2011

%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 4*(2*k+1)^2*x*(1+4*x)^2/(4*(2*k+1)^2*x*(1+4*x)^2 + (k+1)^2*(1+4*x)^2/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 01 2013

%F 0 = +a(n)*(+393216*a(n+2) -119040*a(n+3) +6860*a(n+4)) +a(n+1)*(-16128*a(n+2) +6928*a(n+3) -465*a(n+4)) +a(n+2)*(+36*a(n+2) -63*a(n+3) +6*a(n+4)) for all n in Z. - _Michael Somos_, Aug 06 2014

%F Integral representation as the n-th moment of a positive function W(x) on (0,16), in Maple notation, W(x)=EllipticK(sqrt(1-x/16)/(2*Pi^2*sqrt(x)); a(n)=int(x^n* W(x), x=0..16), n>=0. The function W(x) is singular at x=0 and W(16)=1/(16*Pi). This representation is unique since W(x) is the solution of the Hausdorff moment problem. - Stanley Smith and _Karol A. Penson_, Jun 19 2015

%F a(n) ~ 16^n*(2-2/(8*n+2)^2+21/(8*n+2)^4-671/(8*n+2)^6+45081/(8*n+2)^8)^2/((4*n+1)* Pi). - _Peter Luschny_, Oct 14 2015

%F a(n) = binomial(2*n,n)*binomial(2*n,n) = ( [x^n](1 + x)^(2*n) ) *( [x^n](1 + x)^(2*n) ) = [x^n](F(x)^(4*n)), where F(x) = 1 + x + x^2 + 4*x^3 + 20*x^4 + 120*x^5 + 798*x^6 + 5697*x^7 + ... appears to have integer coefficients. For similar results see A000897, A002897, A006480, A008977, A186420 and A188662. - _Peter Bala_, Jul 14 2016

%F a(n) = Sum_{k = 0..n} binomial(2*n + k,k)*binomial(n,k)^2. Cf. A005258(n) = Sum_{k = 0..n} binomial(n + k,k)*binomial(n,k)^2. - _Peter Bala_, Jul 27 2016

%F a(n) = A241530(2*n), n >= 0. - _Wolfdieter Lang_, Sep 06 2016

%F E.g.f.: 2F2(1/2,1/2; 1,1; 16*x). - _Ilya Gutkovskiy_, Jan 23 2018

%F a(n) = 16^n*hypergeom([1/2, -2*n, 2*n + 1], [1, 1], 1). - _Peter Luschny_, Mar 14 2018

%F The right-hand side of the binomial coefficient identity Sum_{k = 0..n} C(n,k)*C(n+k,k)*C(2*n+2*k,n+k)*(-4)^(n-k) = a(n). - _Peter Bala_, Mar 16 2018

%F a(n) = [x^n] (1 - x)^(2*n) * P(2*n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. Compare with A245086(n) = [x^n] (1 - x)^(2*n) * P(n,(1 + x)/(1 - x)). - _Peter Bala_, Mar 23 2022

%F a(n) = Sum_{k=0..n} multinomial(2n [k k (n-k) (n-k)]), which is another way to count random walks on Z^2, with steps of (0,+-1) or (+-1,0), that return to the point of origin after 2n steps (not necessarily for the first time), as is C(2n,n)^2. - _Shel Kaphan_, Jan 12 2023

%F 0 = a(n)*(+393216*a(n+2) -119040*a(n+3) +6860*a(n+4)) +a(n+1)*(-16128*a(n+2) +6928*a(n+3) -465*a(n+4)) +a(n+2)*(+36*a(n+2) -63*a(n+3) +6*a(n+4)) for n>=0. - _Michael Somos_, May 30 2023

%F From _Peter Bala_, Sep 12 2023: (Start)

%F Right-hand side of the binomial coefficient identities

%F 1) Sum_{k = 0..n} (-1)^(n+k) * C(n,k)*C(n+k,n)*C(2*n+k,n) = a(n).

%F 2) 2*Sum_{k = 0..n} (-1)^(n+k) * C(n,k)*C(n+k-1,n)*C(2*n+k-1,n) = a(n) for n >= 1.

%F 3) (4/3)*Sum_{k = 0..n} (-1)^(n+k) * C(n,k)*C(n+k,n)*C(2*n+k-1,n) = a(n) for n >= 1. (End)

%e G.f. = 1 + 4*x + 36*x^2 + 400*x^3 + 4900*x^4 + 63504*x^5 + 853776*x^6 + ... - _Michael Somos_, Aug 06 2014

%e From _Peter Bala_, Jan 26 2018: (Start)

%e a(2) = 36: The thirty six 3 x k arrays with columns belonging to the set of column vectors S = {[1,0,0], [0,1,0], [1,0,1], [0,1,1]} and having all row sums equal to 2 are the 6 distinct arrays obtained by permuting the columns of

%e /1 1 0 0\

%e |0 0 1 1|,

%e \0 0 1 1/

%e the 6 distinct arrays obtained by permuting the columns of

%e /0 0 1 1\

%e |1 1 0 0|

%e \0 0 1 1/

%e and the 24 arrays obtained by permuting the columns of

%e /1 0 1 0\

%e |0 1 0 1|. (End)

%e \0 0 1 1/

%p A002894 := n-> binomial(2*n,n)^2.

%t CoefficientList[Series[Hypergeometric2F1[1/2, 1/2, 1, 16x], {x, 0, 20}], x]

%t Table[Binomial[2n,n]^2,{n,0,20}] (* _Harvey P. Dale_, Jul 06 2011 *)

%t a[ n_] := SeriesCoefficient[ EllipticK[16 x] / (Pi/2), {x, 0, n}]; (* _Michael Somos_, Aug 06 2014 *)

%t a[n_] := 16^n HypergeometricPFQ[{1/2, -2 n, 2 n + 1}, {1, 1}, 1];

%t Table[a[n], {n, 0, 19}] (* _Peter Luschny_, Mar 14 2018 *)

%o (PARI) {a(n) = binomial(2*n, n)^2};

%o (PARI) {a(n) = if( n<0, 0, polcoeff( polcoeff( polcoeff( 1 / (1 - x * (y + z + 1/y + 1/z)) + x * O(x^(2*n)), 2*n), 0), 0))}; /* _Michael Somos_, Jun 12 2004 */

%o (Sage) [binomial(2*n, n)**2 for n in range(17)] # _Zerinvary Lajos_, Apr 21 2009

%o (Magma) [Binomial(2*n, n)^2: n in [0..20]]; // _Vincenzo Librandi_, Aug 07 2014

%Y Row sums of A069466.

%Y Row 2 of A268367 (even terms).

%Y Equals 4*A060150.

%Y Cf. A000984, A000515, A010370, A054474 (INVERTi transform), A172390, A000897, A002897, A006480, A008977, A186420, A188662, A000894, A241530, A002898 (walks hex lattice).

%K nonn,nice,easy,changed

%O 0,2

%A _N. J. A. Sloane_

%E Edited by _N. J. A. Sloane_, Feb 18 2016