A118931 Triangle, read by rows, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3*k>=0.

1, 1, 1, 1, 2, 1, 8, 1, 20, 1, 40, 40, 1, 70, 280, 1, 112, 1120, 1, 168, 3360, 2240, 1, 240, 8400, 22400, 1, 330, 18480, 123200, 1, 440, 36960, 492800, 246400, 1, 572, 68640, 1601600, 3203200, 1, 728, 120120, 4484480, 22422400, 1, 910, 200200, 11211200, 112112000, 44844800

Offset: 0

Author

Paul D. Hanna, May 06 2006

Keywords

• nonn
• tabl

Comments

Row n contains 1+floor(n/3) terms. Row sums yield A001470. Given column vector V = A118932, then V is invariant under matrix product T*V = V, or, A118932(n) = Sum_{k=0..n} T(n,k)*A118932(k). Given C = Pascal's triangle and T = this triangle, then matrix product M = C^-1*T yields M(3n,n) = (3*n)!/(n!*3^n), 0 otherwise (cf. A100861 formula due to Paul Barry).

Examples

			Triangle T begins:
  1;
  1;
  1;
  1,   2;
  1,   8;
  1,  20;
  1,  40,    40;
  1,  70,   280;
  1, 112,  1120;
  1, 168,  3360,   2240;
  1, 240,  8400,  22400;
  1, 330, 18480, 123200;
  1, 440, 36960, 492800, 246400;
		

Links

• G. C. Greubel, Rows n = 0..150 of the triangle, flattened

Crossrefs

Cf. A001470 (row sums), A118932 (invariant vector).

Variants: A100861, A118933.

Programs

Magma

F:= Factorial;
[n lt 3*k select 0 else F(n)/(3^k*F(k)*F(n-3*k)): k in [0..Floor(n/3)], n in [0..20]]; // G. C. Greubel, Mar 07 2021

Maple

Trow := n -> seq(n!/(j!*(n - 3*j)!*(3^j)), j = 0..n/3):
seq(Trow(n), n = 0..14); # Peter Luschny, Jun 06 2021

Mathematica

T[n_,k_]:= If[n<3*k, 0, n!/(3^k*k!*(n-3*k)!)];
Table[T[n,k], {n,0,20}, {k,0,Floor[n/3]}]//Flatten (* G. C. Greubel, Mar 07 2021 *)

PARI

T(n,k)=if(n<3*k,0,n!/(k!*(n-3*k)!*3^k))

Sage

f=factorial;
flatten([[0 if n<3*k else f(n)/(3^k*f(k)*f(n-3*k)) for k in [0..n/3]] for n in [0..20]]) # G. C. Greubel, Mar 07 2021

Formula

E.g.f.: A(x,y) = exp(x + y*x^3/3).