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%I #11 Oct 22 2022 16:19:40
%S 1,1,-4,5,1,-12,55,-114,94,1,-24,238,-1248,3661,-5736,3828,1,-40,690,
%T -6700,40053,-151060,351800,-465000,270576,1,-60,1595,-24720,247203,
%U -1665900,7660565,-23745720,47560876,-55805520,29400480,1,-84,3185,-72030,1081353,-11344872,85234175,-461800710,1790256286,-4843901664,8693117160,-9320129280,4546558080
%N Coefficient array of numerator polynomials of the ordinary generating functions for the alternating sums of powers for the numbers 1,2,...,2*n+1.
%C The row length sequence of this array is A005408(n), n>=0: 1,3,5,7,...
%C This is the array for the numerator polynomials of the o.g.f. of alternating sums of powers of the first 2*n+1 positive integers.
%C The corresponding array for the first 2*n positive integers is found in A196847.
%C The obvious e.g.f. of a(k,2*n+1) := Sum_{j=1..2*n+1} (-1)^(j+1) * j^k is go(n,x) := Sum_{k>=0} a(k,2*n+1)*(x^k)/k! = Sum_{j=1..2*n+1} (-1)^(j+1) * exp(j*x) = exp(x)*(exp((2*n+1)*x) + 1)/(exp(x) + 1).
%C Via Laplace transformation (see the link under A196837, addendum) one finds the corresponding o.g.f.: Go(n,x) = Po(n,x)/Product_{j=1..2*n+1} (1 - j*x) with the numerator polynomial Po(n,x) = Sum_{m=0..2*n} a(n,m)*x^m.
%F a(n,m) = [x^m](Go(n,x)*Product_{j=1..2*n+1} (1-j*x)), with the o.g.f. Go(n,x) of the sequence a(k,2*n+1) := Sum_{j=1..2*n+1} (-1)^(j+1) * j^k. See a comment above.
%F a(n,0) = 1, n >= 0, and a(n,m) = (-1)^m*((Sum_{i=1..n} S_{2*i-1,2*i}(2*(n-1),m)) + |s(2*n+1,2n+1-m)|), n >= 0, m = 1..2*n, with the (i,j)-family of number triangles S_{i,j}(n,k) defined in a comment on A196845, and the Stirling numbers of the first kind s(n,m) = A048994(n,m).
%e n\m 0 1 2 3 4 5 6 7 8
%e 0: 1
%e 1: 1 -4 5
%e 2: 1 -12 55 -114 94
%e 3: 1 -24 238 -1248 3661 -5736 3828
%e 4: 1 -40 690 -6700 40053 -151060 351800 -465000, 270576
%e ...
%e The o.g.f. for the sequence a(k,5) := (1^k - 2^k + 3^k - 4^k + 5^k) = A198628(k), k >= 0, (n=2) is Go(2,x) = (1 - 12*x + 55*x^2 - 114*x^3 + 94*x^4)/Product_{j=1..5} (1-j*x).
%e a(3,2) = S_{1,2}(5,1) + S_{3,4}(5,1) + S_{5,6}(5,1) + |s(7,5)| = A196845(5,1) + A196846(5,1) + 17 + |s(7,5)| = 25+21+17+175 = 238. Here S_{5,6}(5,1) = 1+2+3+4+7 = 17 was used.
%Y Cf. A196847, A196837.
%K sign,easy,tabf
%O 0,3
%A _Wolfdieter Lang_, Oct 27 2011
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