A074052 The lowest order term in an expansion of Sum_{i=1..m} i^n*(i+1)! in a special factorial basis.

0, -2, 2, 2, -14, 26, 34, -398, 1210, 450, -23406, 118634, -166286, -1983342, 18159658, -68002894, -112926670, 3497644570, -24969255550, 64943618962, 607880756218, -9318511004702, 60525142971954, -80108659182870, -3000122066181358

Offset: 0

Author

Jan Fricke, Aug 14 2002

Keywords

• easy
• sign

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).

Examples

			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)!.
		

Crossrefs

Cf. A074051, A197184.

Programs

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] ]

Extensions

More terms from R. J. Mathar, Oct 11 2011