A111887 Seventh column of triangle A112492 (inverse scaled Pochhammer symbols).

1, 13068, 104587344, 673781602752, 3878864920694016, 21006340945438768128, 110019668725577574273024, 565858042127972959667208192, 2882220940619488483325345857536, 14605752814655604919042956624396288

Offset: 0

Author

Wolfdieter Lang, Sep 12 2005

Keywords

• nonn
• easy

Comments

Also continuation of family of differences of reciprocals of unity. See A001242, A111886 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 6 in triangle A008969.

Cf. A001242, A111886, A112492.

Programs

Magma

A111887:= func< n | (-1)*Factorial(7)^n*(&+[(-1)^j*Binomial(7,j)/j^n : j in [1..7]]) >;
[A111887(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+6,6], {n,0,30}] (* G. C. Greubel, Jul 24 2023 *)

PARI

a(n) = -((7!)^n)*sum(j=1, 7, ((-1)^j)*binomial(7, 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 A111887(n): return T(n+6,6)
[A111887(n) for n in range(31)] # G. C. Greubel, Jul 24 2023

Formula

G.f.: 1/Product_{j=1..7} 1-7!*x/j.

a(n) = -((7!)^n) * Sum_{j=1..7} (-1)^j*binomial(7, j)/j^n, n>=0.

a(n) = A112492(n+6, 7), n>=0.