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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.

%I A074052 #15 Jan 07 2025 15:16:56
%S A074052 0,-2,2,2,-14,26,34,-398,1210,450,-23406,118634,-166286,-1983342,
%T A074052 18159658,-68002894,-112926670,3497644570,-24969255550,64943618962,
%U A074052 607880756218,-9318511004702,60525142971954,-80108659182870,-3000122066181358
%N A074052 The lowest order term in an expansion of Sum_{i=1..m} i^n*(i+1)! in a special factorial basis.
%C A074052 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).
%e A074052 a(0) = 0 because Sum_{i=1..m} (i+1)! = 0 + 1*Sum_{i=1..m} (i+1)! + 0*(m+2)!.
%e A074052 a(1) = -2 because Sum_{i=1..m} i*(i+1)! = -2 -1*Sum_{i=1..m} (i+1)! + 1*(m+2)!.
%e A074052 a(2) = 2 because Sum_{i=1..m} i^2*(i+1)! = 2 +0*Sum_{i=1..m} (i+1)! + (m-1)*(m+2)!.
%e A074052 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)!.
%e A074052 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)!.
%t A074052 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] ]
%Y A074052 Cf. A074051, A197184.
%K A074052 easy,sign
%O A074052 0,2
%A A074052 _Jan Fricke_, Aug 14 2002
%E A074052 More terms from R. J. Mathar, Oct 11 2011