OFFSET
0,2
COMMENTS
For each n there unique 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(n), and this sequence here are the a(n).
EXAMPLE
a(0) = 0 because sum_{i=1..m} (i+1)! = 0 + 1*Sum_{i=1..m} (i+1)! + 0*(m+2)!.
a(1) = -2 because sum_{i=1..m} i*(i+1)! = -2 -1*sum_{i=1..m} (i+1)! +1*(m+2)!.
a(2) = 2 because sum_{i=1..m} i^2*(i+1)! = 2 +0*sum_{i=1..m} (i+1)!+ (m-1)*(m+2)!.
a(3) = 2 because Sum_{i=1..n} i^3*(i+1)! = 2 +3*sum_{i=1..m} (i+1)!+(m^2-m-1)*(m+2)!.
a(4)=-14 because sum_{i=1..n}i^4*(i+1)! = -14 -7*Sum_{i=1..n} (i+1)! +(m^3-m^2-2*m+7)* (m+2)!.
MATHEMATICA
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] ]
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Jan Fricke, Aug 14 2002
EXTENSIONS
More terms from R. J. Mathar, Oct 11 2011
STATUS
approved