0,4

If a(n)=0 then Sum_{i=1..infty}i^n(i+1)! = b(n) in the p-adic numbers. The only known numbers n with a(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)*Sum_{i=1..m}(i+1)! + p_n(m)(m+2)! + b(n) The sequence b(n) is A074052.

Second inverse binomial transform of A000587. E.g.f.: exp(1-2*x-exp(-x)). G.f.: Sum((x/(1+2*x))^k/Product(1+l*x/(1+2*x), l = 0 .. k), k = 0 .. infinity)/(1+2*x). a(n) = Sum_{k=0..n} (-1)^(n-k)*(k^2-3*k+1)*Stirling2(n, k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 27 2005

a(2)=0 because Sum_{i=1..m}i^2(i+1)! = (m-1)(m+2)!+2. a(3)=3 because Sum_{i=1..m}i^3(i+1)! = 3*Sum_{i=1..m}(i+1)!+(m^2-m-1)(m+2)!+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)]] ]; Expand[n^a - (n + 2)p[n] + p[n - 1]] ]

Cf. A074052.

Sequence in context: A134976 A010622 A103844 this_sequence A048292 A072450 A085785

Adjacent sequences: A074048 A074049 A074050 this_sequence A074052 A074053 A074054

easy,sign

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

More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 27 2005