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A196847 Coefficient table of numerator polynomials of the ordinary generating function for the alternating power sums for the numbers 1,2,...,2*n.

%I A196847 #11 Oct 22 2022 16:20:03
%S A196847 1,1,-5,7,1,-14,73,-168,148,1,-27,298,-1719,5473,-9162,6396,1,-44,830,
%T A196847 -8756,56453,-227744,562060,-778800,468576,1,-65,1865,-31070,332463,
%U A196847 -2385305,11612795,-37875240,79269676,-96420480,52148160,1,-90,3647,-87900,140202
%N A196847 Coefficient table of numerator polynomials of the ordinary generating function for the alternating power sums for the numbers 1,2,...,2*n.
%C A196847 The row length sequence of this array is A005408(n-1), n >= 1: 1,3,5,7,...
%C A196847 This is the array for the numerator polynomials of the o.g.f. of alternating power sums of the first 2*n positive integers.
%C A196847 The corresponding array for the first 2*n+1 positive integers is found in A196848.
%C A196847 The obvious e.g.f. of a(k,2*n) := Sum_{j=1..2*n} (-1)^j * j^k is ge(n,x) := Sum_{k>=0} a(k,2*n)*(x^k)/k! = Sum_{j=1..2*n} (-1)^j * exp(j*x) = exp(x)*(exp(2*n*x) - 1)/(exp(x) + 1).
%C A196847 Via Laplace transformation (see the link under A196837, addendum) one finds the corresponding o.g.f.: Ge(n,x) = n*x*Pe(n,x)/Product_{j=1..2*n} (1 - j*x) with the numerator polynomial Pe(n,x) = Sum_{m=0..2*(n-1)} a(n,m)*x^m.
%F A196847 a(n,m) = [x^m](Ge(n,x)*Product_{j=1..2*n} (1 - j*x/(n*x))), with the o.g.f. Ge(n,x) of the sequence a(k,2*n) := Sum_{j=1..2*n} (-1)^j * j^k. See a comment above.
%F A196847 a(n,m) = (1/n)*(-1)^m*Sum_{i=1..n} S_{2*i-1,2*i}(2*(n-1),m), n >= 1, with the (i,j)-family of number triangles S_{i,j}(n,k) defined in a comment to A196845.
%e A196847 n\m 0   1   2     3     4       5      6       7      8
%e A196847 1:  1
%e A196847 2:  1  -5   7
%e A196847 3:  1 -14  73  -168   148
%e A196847 4:  1 -27 298 -1719  5473   -9162   6396
%e A196847 5:  1 -44 830 -8756 56453 -227744 562060 -778800 468576
%e A196847 ...
%e A196847 The o.g.f. for the sequence a(k,4) := -(1^k - 2^k + 3^k -4^k) = 2*A053154(k), k>=0, (n=2) is Ge(2,x) = 2*x*(1-5*x+7*x^2)/Product_{j=1..4} (1 - j*x).
%e A196847 a(3,2) = (S_{1,2}(4,2) + S_{3,4}(4,2) + S_{5,6}(4,2))/3 = (A196845(4,2) + A196846(4,2) + |s(5,3)|)/3 = (119+65+35)/3 = 73. Here S_{5,6}(4,2) = a_2(1,2,3,4) = |s(5,3)|, with the Stirling numbers of the first kind s(n,m) = A048994(n,m) was used.
%Y A196847 Cf. A196848, A196837.
%K A196847 sign,easy,tabf
%O A196847 1,3
%A A196847 _Wolfdieter Lang_, Oct 27 2011