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A074052
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The lowest order term in an expansion of sum_{i=1..m}*i^n*(i+1)! in a special factorial basis.
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2
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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
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OFFSET
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0,2
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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)!.
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MATHEMATICA
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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] ]
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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EXTENSIONS
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More terms from R. J. Mathar, Oct 11 2011
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STATUS
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approved
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