A111888 Eighth column of triangle A112492 (inverse scaled Pochhammer symbols).

1, 109584, 7245893376, 381495483224064, 17810567950611972096, 778101042571221893382144, 32762625292956765972873609216, 1351813956241264848815287984717824

Offset: 0

Author

Wolfdieter Lang, Sep 12 2005

Keywords

• nonn
• easy

Comments

Also continuation of family of Differences of reciprocals of unity. See A001242, A111887 and triangle A008969.

Links

• Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.

Crossrefs

Also right-hand column 7 in triangle A008969.

Cf. A001242, A111887, A112492.

Programs

Magma

A111888:= func< n | (-1)*Factorial(8)^n*(&+[(-1)^j*Binomial(8,j)/j^n : j in [1..8]]) >;
[A111888(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+7,7], {n,0,30}] (* G. C. Greubel, Jul 24 2023 *)

PARI

a(n) = -((8!)^n)*sum(j=1, 8, ((-1)^j)*binomial(8, 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 A111888(n): return T(n+7,7)
[A111888(n) for n in range(31)] # G. C. Greubel, Jul 24 2023

Formula

G.f.: 1/Product_{j=1..8} 1-8!*x/j.

a(n) = -((8!)^n) * Sum_{j=1..8} (-1)^j*binomial(8, j)/j^n, n>=0.

a(n) = A112492(n+7, 8), n>=0.