%I #68 Nov 12 2024 18:44:08

%S 1,3,1,3,4,1,3,7,5,1,3,10,12,6,1,3,13,22,18,7,1,3,16,35,40,25,8,1,3,

%T 19,51,75,65,33,9,1,3,22,70,126,140,98,42,10,1,3,25,92,196,266,238,

%U 140,52,11,1,3,28,117,288,462,504,378,192,63,12,1,3,31,145,405,750,966,882,570,255,75,13,1

%N (3,1) Pascal triangle.

%C The array F(3;n,m) gives in the columns m >= 1 the figurate numbers based on A016777, including the pentagonal numbers A000326 (see the W. Lang link).

%C This is the third member, d=3, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal (d=1), A029653 (d=2).

%C This is an example of a Riordan triangle (see A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group) with o.g.f. of column nr. m of the type g(x)*(x*f(x))^m with f(0)=1. Therefore the o.g.f. for the row polynomials p(n,x):=Sum_{m=0..n} a(n,m)*x^m is G(z,x)=g(z)/(1-x*z*f(z)). Here: g(x)=(1+2*x)/(1-x), f(x)=1/(1-x), hence G(z,x)=(1+2*z)/(1-(1+x)*z).

%C The SW-NE diagonals give the Lucas numbers A000032: L(n) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with L(0)=2. Observation by _Paul Barry_, Apr 29 2004. Proof via recursion relations and comparison of inputs.

%C Triangle T(n,k), read by rows, given by [3,-2,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Sep 17 2009

%C For a closed-form formula for generalized Pascal's triangle see A228576. - _Boris Putievskiy_, Sep 09 2013

%C From _Wolfdieter Lang_, Jan 09 2015: (Start)

%C The signed lower triangular matrix (-1)^(n-1)*a(n,m) is the inverse of the Riordan matrix A106516; that is Riordan ((1-2*x)/(1+x),x/(1+x)).

%C See the _Peter Bala_ comment from Dec 23 2014 in A106516 for general Riordan triangles of the type (g(x), x/(1-x)): exp(x)*r(n,x) = d(n,x) with the e.g.f. r(n,x) of row n and the e.g.f. of diagonal n.

%C Similarly, for general Riordan triangles of the type (g(x), x/(1+x)): exp(x)*r(n,-x) = d(n,x). (End)

%C The n-th row polynomial is (3 + x)*(1 + x)^(n-1) for n >= 1. More generally, the n-th row polynomial of the Riordan array ( (1-a*x)/(1-b*x), x/(1-b*x) ) is (b - a + x)*(b + x)^(n-1) for n >= 1. - _Peter Bala_, Mar 02 2018

%C Binomial(n-2,k)+2*Binomial(n-3,k) is also the number of permutations avoiding both 123 and 132 with k double descents, i.e., positions with w[i]>w[i+1]>w[i+2]. - _Lara Pudwell_, Dec 19 2018

%D Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.

%D Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.

%H Reinhard Zumkeller, <a href="/A093560/b093560.txt">Rows n = 0..125 of triangle, flattened</a>

%H P. Bala, <a href="/A081577/a081577.pdf">A note on the diagonals of a proper Riordan Array</a>

%H M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, <a href="https://arxiv.org/abs/1812.07112">Distributions of Statistics over Pattern-Avoiding Permutations</a>, arXiv preprint arXiv:1812.07112 [math.CO], 2018.

%H W. Lang, <a href="/A093560/a093560.txt">First 10 rows and array of figurate numbers </a>.

%F a(n, m)=F(3;n-m, m) for 0<= m <= n, otherwise 0, with F(3;0, 0)=1, F(3;n, 0)=3 if n>=1 and F(3;n, m):=(3*n+m)*binomial(n+m-1, m-1)/m if m>=1.

%F G.f. column m (without leading zeros): (1+2*x)/(1-x)^(m+1), m>=0.

%F Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=3 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).

%F T(n, k) = C(n, k) + 2*C(n-1, k). - _Philippe Deléham_, Aug 28 2005

%F Equals M * A007318, where M = an infinite triangular matrix with all 1's in the main diagonal and all 2's in the subdiagonal. - _Gary W. Adamson_, Dec 01 2007

%F Sum_{k=0..n} T(n,k) = A151821(n+1). - _Philippe Deléham_, Sep 17 2009

%F exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(3 + 7*x + 5*x^2/2! + x^3/3!) = 3 + 10*x + 22*x^2/2! + 40*x^3/3! + 65*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - _Peter Bala_, Dec 22 2014

%F G.f.: (-1-2*x)/(-1+x+x*y). - _R. J. Mathar_, Aug 11 2015

%e Triangle begins

%e 1,

%e 3, 1,

%e 3, 4, 1,

%e 3, 7, 5, 1,

%e 3, 10, 12, 6, 1,

%e 3, 13, 22, 18, 7, 1,

%e 3, 16, 35, 40, 25, 8, 1,

%e 3, 19, 51, 75, 65, 33, 9, 1,

%e 3, 22, 70, 126, 140, 98, 42, 10, 1,

%e 3, 25, 92, 196, 266, 238, 140, 52, 11, 1,

%o (Haskell)

%o a093560 n k = a093560_tabl !! n !! k

%o a093560_row n = a093560_tabl !! n

%o a093560_tabl = [1] : iterate

%o (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [3, 1]

%o -- _Reinhard Zumkeller_, Aug 31 2014

%o (GAP) Concatenation([1],Flat(List([1..11],n->List([0..n],k->Binomial(n,k)+2*Binomial(n-1,k))))); # _Muniru A Asiru_, Dec 20 2018

%o (Python)

%o from math import comb, isqrt

%o def A093560(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),a:=n-comb(r+1,2))*(r+(r-a<<1))//r if n else 1 # _Chai Wah Wu_, Nov 12 2024

%Y Cf. Column sequences for m=1..9: A016777, A000326 (pentagonal), A002411, A001296, A051836, A051923, A050494, A053367, A053310;

%Y A007318 (Pascal's triangle), A029653 ((2,1) Pascal triangle), A093561 ((4,1) Pascal triangle), A228196, A228576.

%K nonn,tabl,easy

%O 0,2

%A _Wolfdieter Lang_, Apr 22 2004

%E Incorrect connection with A046055 deleted by _N. J. A. Sloane_, Jul 08 2009