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 A136216 Triangle T, read by rows, where T(n,k) = A008544(n-k)*C(n,k) where A008544 equals the triple factorials in column 0. 4
 1, 2, 1, 10, 4, 1, 80, 30, 6, 1, 880, 320, 60, 8, 1, 12320, 4400, 800, 100, 10, 1, 209440, 73920, 13200, 1600, 150, 12, 1, 4188800, 1466080, 258720, 30800, 2800, 210, 14, 1, 96342400, 33510400, 5864320, 689920, 61600, 4480, 280, 16, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 The row polynomials form an Appell sequence. - Tom Copeland, Dec 03 2013 LINKS Wikipedia, Appell sequence Wikipedia, Sheffer sequence FORMULA Column k of T = column 0 of V^(k+1) for k>=0 where V = A112333. Equals the matrix square of triangle A136215. T(n,k) = (3*n-3*k-1)*T(n-1,k) + T(n-1,k-1). - Peter Bala, Jul 10 2008 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 From Peter Bala, Aug 28 2013: (Start) Exponential Riordan array [1/(1 - 3*y)^(2/3), y]. 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). 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) EXAMPLE Triangle begins: 1; 2, 1; 10, 4, 1; 80, 30, 6, 1; 880, 320, 60, 8, 1; 12320, 4400, 800, 100, 10, 1; 209440, 73920, 13200, 1600, 150, 12, 1; 4188800, 1466080, 258720, 30800, 2800, 210, 14, 1; ... MATHEMATICA (* The function RiordanArray is defined in A256893. *) RiordanArray[1/(1 - 3 #)^(2/3)&, #&, 9, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *) PROG (PARI) {T(n, k) = binomial(n, k)*if(n-k==0, 1, prod(j=0, n-k-1, 3*j+2))} for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) CROSSREFS Cf. A136215 (square-root), A112333, A008544, A136212, A136213. Cf. A094587. Sequence in context: A110682 A110327 A105615 * A121334 A126450 A235608 Adjacent sequences:  A136213 A136214 A136215 * A136217 A136218 A136219 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Feb 07 2008 STATUS approved

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Last modified October 21 01:18 EDT 2019. Contains 328291 sequences. (Running on oeis4.)