A111886 Sixth column of triangle A112492 (inverse scaled Pochhammer symbols).

1, 1764, 1942416, 1744835904, 1413470290176, 1083688832185344, 806595068762689536, 590914962115587293184, 429295503918929370218496, 310518802877016005311463424, 224098118280955193084850733056

Offset: 0

Author

Wolfdieter Lang, Sep 12 2005

Keywords

• nonn
• easy

Comments

Also continuation of family of differences of reciprocals of unity. See A001242 and triangle A008969.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250
• Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.

Crossrefs

Also right-hand column 5 in triangle A008969.

Programs

Magma

A111886:= func< n | (-1)*Factorial(6)^n*(&+[(-1)^j*Binomial(6,j)/j^n : j in [1..6]]) >;
[A111886(n): n in [0..30]]; // G. C. Greubel, Jul 24 2023

Mathematica

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, (k+1)^(n-k)*T[n-1,k-1] +k!*T[n-1,k]]; (* T = A112492 *)
Table[T[n+5,5], {n,0,30}] (* G. C. Greubel, Jul 24 2023 *)

PARI

a(n) = -((6!)^n)*sum(j=1, 6, (-1)^j*binomial(6, j)/j^n); \\ Michel Marcus, Apr 28 2020

SageMath

@CachedFunction
def T(n,k): # T = A112492
    if (k==0 or k==n): return 1
    else: return (k+1)^(n-k)*T(n-1,k-1) + factorial(k)*T(n-1,k)
def A111886(n): return T(n+5,5)
[A111886(n) for n in range(31)] # G. C. Greubel, Jul 24 2023

Formula

G.f.: 1/Product_{j=1..6} (1-6!*x/j).

a(n) = -((6!)^n)*Sum_{j=1..6} (-1)^j*binomial(6, j)/j^n, n >= 0.

a(n) = A112492(n+5, 6), n>=0.