OFFSET
0,3
COMMENTS
Hankel transform is 2^n. Successive binomial transforms are A002426, A000984, A026375, A081671, A098409, A098410.
From Andrew V. Sutherland, Feb 29 2008: (Start)
Counts returning walks of length n on a 1-d integer lattice with step set {-1,+1}.
Moment sequence of the trace of a random matrix in G = SO(2). If X = tr(A) is a random variable (A distributed with Haar measure on G), then a(n) = E[X^n].
Also the moment sequence of the trace of the k-th power of a random matrix in USp(2) = SU(2), for all k > 2.
(End)
From Paul Barry, Aug 10 2009: (Start)
The Hankel transform of 0,1,0,2,0,6,... is 0,-1,0,4,0,-16,0,... with general term I*(-4)^(n/2)(1 - (-1)^n)/4, I = sqrt(-1).
The Hankel transform of 1,1,0,2,0,6,... (which has g.f. 1 + x/sqrt(1 - 4*x^2)) is A164111. (End)
a(n) is the total number of closed walks (round trips) of length n on the graph P_N (a line with N nodes and N-1 edges), divided by N, in the limit N -> infinity. See a comment on A198632 and a link under A201198. - Wolfdieter Lang, Oct 10 2012
REFERENCES
Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011.
Francesc Fite and Andrew V. Sutherland, Sato-Tate distributions of twists of y^2= x^5-x and y^2= x^6+1, arXiv preprint arXiv:1203.1476 [math.NT], 2012. - From N. J. A. Sloane, Sep 14 2012
Nikita Gogin and Mika Hirvensalo, On the Moments of Squared Binomial Coefficients, (2020).
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
FORMULA
From Andrew V. Sutherland, Feb 29 2008: (Start)
a(2*n) = binomial(2*n,n) = A000984(n); a(2*n+1) = 0.
a(n) = Sum_{k = 0..n} A107430(n,k)*(-1)^(n-k).
a(n) = Sum_{k = 0..n} A061554(n,k)*(-1)^k.
a(n) = (1/Pi)*Integral_{t = 0..Pi} cos^n(t) dt. (End)
E.g.f.: I_0 (2x) where I_n(x) is the modified Bessel function as a function of x. - Benjamin Phillabaum, Mar 10 2011
G.f.: A(x) = 1/sqrt(1 - 4*x^2) - Vladimir Kruchinin, Apr 16 2011
a(n) = (1/Pi)*Integral{x = -2..2} x^n/sqrt((2 - x)*(2 + x)). - Peter Luschny, Sep 12 2011
a(n) = (-1)^floor(n/2) Hypergeometric([-n,-n],[1], -1). - Peter Luschny, Nov 01 2011
E.g.f.: E(0)/(1 - x) where E(k) = 1 - x/(1 - x/(x - (k+1)^2/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013
E.g.f.: 1 + x^2/(Q(0) - x^2), where Q(k)= x^2 + (k+1)^2 - x^2*(k+1)^2/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 28 2013
G.f.: 1/(1 - 2*x^2*Q(0)), where Q(k)= 1 + (4*k+1)*x^2/(k+1 - x^2*(2*k+2)*(4*k+3)/(2*x^2*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + (k+1)/(x*(2*k+1))/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
G.f.: G(0)/(1+x), where G(k) = 1 + x*(2+5*x)*(4*k+1)/((4*k+2)*(1+x)^2 - 2*(2*k+1)*(4*k+3)*x*(2+5*x)*(1+x)^2/((4*k+3)*x*(2+5*x) + 4*(k+1)*(1+x)^2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 19 2014
a(n) = 2^n*JacobiP(n,0,-1/2-n,-3)). - Peter Luschny, Aug 02 2014
a(n) = (2^(n-1)*((-1)^n+1)*Gamma((n+1)/2))/(sqrt(Pi)*Gamma((n+2)/2)). - Peter Luschny, Sep 10 2014
a(n) = n!*[x^n]hypergeom([],[1],x^2). - Peter Luschny, Jan 31 2015
a(n) = 2^n*hypergeom([1/2,-n],[1],2). - Peter Luschny, Feb 03 2015
From Peter Bala, Jul 25 2016: (Start)
a(n) = (-1)^floor(n/2)*Sum_{k = 0..n} (-1)^k*binomial(n,k)^2.
D-finite with recurrence: a(n) = 4*(n - 1)/n * a(n-2) with a(0) = 1, a(1) = 0. (End)
From Ilya Gutkovskiy, Jul 25 2016: (Start)
Inverse binomial transform of A002426.
a(n) = Sum_{k=0..n} (-1)^k*A128014(k).
a(n) ~ 2^n*((-1)^n + 1)/sqrt(2*Pi*n). (End)
EXAMPLE
a(4) = 6 {UUDD,UDUD,UDDU,DUUD,DUDU,DDUU}.
MAPLE
seq((-1)^(n/2)*pochhammer(-n, n/2)/(n/2)!, n=0..43); # Peter Luschny, May 17 2013
seq(n!*coeff(series(hypergeom([], [1], x^2), x, n+1), x, n), n=0..42); # Peter Luschny, Jan 31 2015
MATHEMATICA
Table[(-1)^Floor[n/2] HypergeometricPFQ[{-n, -n}, {1}, -1], {n, 0, 30}] (* Peter Luschny, Nov 01 2011 *)
PROG
(Haskell)
a126869 n = a204293_row (2*n) !! n -- Reinhard Zumkeller, Jan 14 2012
(Sage)
A126869 = lambda n: (2^(n-1)*((-1)^n+1)*gamma((n+1)/2))/(sqrt(pi)*gamma((n+2)/2))
[A126869(n) for n in range(44)] # Peter Luschny, Sep 10 2014
CROSSREFS
This is A000984 with interspersed zeros. m-th binomial transforms of A000984: A126869 (m = -2), A002426 (m = -1 and m = -3 for signed version), A000984 (m = 0 and m = -4 for signed version), A026375 (m = 1 and m = -5 for signed version), A081671 (m = 2 and m = -6 for signed version), A098409 (m = 3 and m = -7 for signed version), A098410 (m = 4 and m = -8 for signed version), A104454 (m = 5 and m = -9 for signed version).
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Mar 16 2007
STATUS
approved