A118933 Triangle, read by rows, where T(n,k) = n!/(k!*(n-4*k)!*4^k) for n>=4*k>=0.

1, 1, 1, 1, 1, 6, 1, 30, 1, 90, 1, 210, 1, 420, 1260, 1, 756, 11340, 1, 1260, 56700, 1, 1980, 207900, 1, 2970, 623700, 1247400, 1, 4290, 1621620, 16216200, 1, 6006, 3783780, 113513400, 1, 8190, 8108100, 567567000, 1, 10920, 16216200, 2270268000, 3405402000

Offset

0,6

Comments

Row n contains 1+floor(n/4) terms. Row sums yield A118934. Given column vector V = A118935, then V is invariant under matrix product T*V = V, or, A118935(n) = Sum_{k=0..n} T(n,k)*A118935(k). Given C = Pascal's triangle and T = this triangle, then matrix product M = C^-1*T yields M(4n,n) = (4*n)!/(n!*4^n), 0 otherwise (cf. A100861 formula due to Paul Barry).

Links

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

Formula

E.g.f.: A(x,y) = exp(x + y*x^4/4).

Example

Triangle begins:
  1;
  1;
  1;
  1;
  1,    6;
  1,   30;
  1,   90;
  1,  210;
  1,  420,   1260;
  1,  756,  11340;
  1, 1260,  56700;
  1, 1980, 207900;
  1, 2970, 623700, 1247400; ...

Mathematica

T[n_, k_]:= If[n<4*k, 0, n!/(4^k*k!*(n-4*k)!)];
Table[T[n, k], {n, 0, 20}, {k, 0, n/4}]//Flatten (* G. C. Greubel, Mar 07 2021 *)

Prog

(PARI) T(n, k)=if(n<4*k, 0, n!/(k!*(n-4*k)!*4^k))
(Sage)
f=factorial;
flatten([[0 if n<4*k else f(n)/(4^k*f(k)*f(n-4*k)) for k in [0..n/4]] for n in [0..20]]) # G. C. Greubel, Mar 07 2021
(Magma)
F:= Factorial;
[n lt 4*k select 0 else F(n)/(4^k*F(k)*F(n-4*k)): k in [0..Floor(n/4)], n in [0..20]]; // G. C. Greubel, Mar 07 2021

Crossrefs

Cf. A118934 (row sums), A118935 (invariant vector).

Variants: A100861, A118931.

Sequence in context: A145629 A193633 A176289 * A046212 A120105 A120101

Adjacent sequences: A118930 A118931 A118932 * A118934 A118935 A118936

Keyword

nonn,tabl

Author

Paul D. Hanna, May 06 2006

Status

approved