0,2

If b(n)=0 then Sum_{i=1..infty}i^n(i+1)! = a(n) in the p-adic numbers. The only known numbers n with b(n)=0 are 2 and 5.

For each n there uniquely determined numbers a(n) and b(n) and a polynomial p_n such that for all integers m: Sum_{i=1..m}i^n(i+1)! = a(n) + b(n)*Sum_{i=1..m}(i+1)! + p_n(m)(m+2)! The sequence b(n) is A074051

a(3)=2 because Sum_{i=1..n}i^3(i+1)! = 2+3*Sum_{i=1..n}(i+1)!+(n^2-n-1)(n+2)! a(4)=-14 because Sum_{i=1..n}i^4(i+1)! = -14+3*Sum_{i=1..n}(i+1)!+(...)(n+2)!

A[a_] := Module[{p, k}, p[n_] = 0; For[k = a - 1, k >= 0, k--, p[n_] = Expand[p[n] + n^k Coefficient[n^a - (n + 2)p[n] + p[n - 1], n^(k + 1)]] ]; -2 p[0] ]

Cf. A074051.

Sequence in context: A077991 A049148 A063898 this_sequence A129409 A025521 A068218

Adjacent sequences: A074049 A074050 A074051 this_sequence A074053 A074054 A074055

easy,sign,uned

Jan Fricke (fricke(AT)uni-greifswald.de), Aug 14 2002