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%I #11 Oct 22 2022 16:20:03
%S 1,1,-5,7,1,-14,73,-168,148,1,-27,298,-1719,5473,-9162,6396,1,-44,830,
%T -8756,56453,-227744,562060,-778800,468576,1,-65,1865,-31070,332463,
%U -2385305,11612795,-37875240,79269676,-96420480,52148160,1,-90,3647,-87900,140202
%N Coefficient table of numerator polynomials of the ordinary generating function for the alternating power sums for the numbers 1,2,...,2*n.
%C The row length sequence of this array is A005408(n-1), n >= 1: 1,3,5,7,...
%C 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 The corresponding array for the first 2*n+1 positive integers is found in A196848.
%C 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 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 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 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 n\m 0 1 2 3 4 5 6 7 8
%e 1: 1
%e 2: 1 -5 7
%e 3: 1 -14 73 -168 148
%e 4: 1 -27 298 -1719 5473 -9162 6396
%e 5: 1 -44 830 -8756 56453 -227744 562060 -778800 468576
%e ...
%e 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 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 Cf. A196848, A196837.
%K sign,easy,tabf
%O 1,3
%A _Wolfdieter Lang_, Oct 27 2011
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