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A074052 The lowest order term in an expansion of sum_{i=1..m}*i^n*(i+1)! in a special factorial basis. 2
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 (list; graph; refs; listen; history; text; internal format)
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).

LINKS

Table of n, a(n) for n=0..24.

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

Cf. A074051, A197184.

Sequence in context: A183584 A289068 A063898 * A129409 A025521 A068218

Adjacent sequences:  A074049 A074050 A074051 * A074053 A074054 A074055

KEYWORD

easy,sign

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified November 18 09:09 EST 2017. Contains 294879 sequences.