A136216 - OEIS

%I #27 Jul 19 2019 14:26:48

%S 1,2,1,10,4,1,80,30,6,1,880,320,60,8,1,12320,4400,800,100,10,1,209440,

%T 73920,13200,1600,150,12,1,4188800,1466080,258720,30800,2800,210,14,1,

%U 96342400,33510400,5864320,689920,61600,4480,280,16,1

%N Triangle T, read by rows, where T(n,k) = A008544(n-k)*C(n,k) where A008544 equals the triple factorials in column 0.

%C This array is the particular case P(2,3) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown in the comments to A094587. - _Peter Bala_, Jul 10 2008

%C The row polynomials form an Appell sequence. - _Tom Copeland_, Dec 03 2013

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Appell_sequence">Appell sequence</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Sheffer_sequence">Sheffer sequence</a>

%F Column k of T = column 0 of V^(k+1) for k>=0 where V = A112333.

%F Equals the matrix square of triangle A136215.

%F T(n,k) = (3*n-3*k-1)*T(n-1,k) + T(n-1,k-1). - _Peter Bala_, Jul 10 2008

%F Using the formalism of A132382 modified for the triple rather than the double factorial (replace 2 by 3 in basic formulas), the e.g.f. for the row polynomials is exp(x*t)*(1-3x)^(-2/3). - _Tom Copeland_, Aug 18 2008

%F From _Peter Bala_, Aug 28 2013: (Start)

%F Exponential Riordan array [1/(1 - 3*y)^(2/3), y].

%F The row polynomials R(n,x) thus form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = sum {k = 0..n} binomial(n,k)*y^(n-k)*R(k,x).

%F Define a polynomial sequence P(n,x) of binomial type by setting P(n,x) = product {k = 0..n-1} (2*x + 3*k) with the convention that P(0,x) = 1. Then this is triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x). For example, row 3 is (80, 30, 6, 1) so P(3,x + 1) = (2*x + 2)*(2*x + 5)*(2*x + 8) = 80 + 20*(2*x) + 6*(2*x*(2*x + 3)) + (2*x)*(2*x + 3)*(2*x + 6). (End)

%e Triangle begins:

%e 1;

%e 2, 1;

%e 10, 4, 1;

%e 80, 30, 6, 1;

%e 880, 320, 60, 8, 1;

%e 12320, 4400, 800, 100, 10, 1;

%e 209440, 73920, 13200, 1600, 150, 12, 1;

%e 4188800, 1466080, 258720, 30800, 2800, 210, 14, 1; ...

%t (* The function RiordanArray is defined in A256893. *)

%t RiordanArray[1/(1 - 3 #)^(2/3)&, #&, 9, True] // Flatten (* _Jean-François Alcover_, Jul 19 2019 *)

%o (PARI) {T(n,k) = binomial(n,k)*if(n-k==0,1, prod(j=0,n-k-1,3*j+2))}

%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

%Y Cf. A136215 (square-root), A112333, A008544, A136212, A136213.

%Y Cf. A094587.

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Feb 07 2008