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 A196848 Coefficient array of numerator polynomials of the ordinary generating functions for the alternating sums of powers for the numbers 1,2,...,2*n+1. 4
 1, 1, -4, 5, 1, -12, 55, -114, 94, 1, -24, 238, -1248, 3661, -5736, 3828, 1, -40, 690, -6700, 40053, -151060, 351800, -465000, 270576, 1, -60, 1595, -24720, 247203, -1665900, 7660565, -23745720, 47560876, -55805520, 29400480, 1, -84, 3185, -72030, 1081353, -11344872, 85234175, -461800710, 1790256286, -4843901664, 8693117160, -9320129280, 4546558080 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The row length sequence of this array is A005408(n), n>=0: 1,3,5,7,... 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. The corresponding array for the first 2*n positive integers is found in A196847. The obvious e.g.f. of a(k,2*n+1) := sum(((-1)^(j+1))*j^k,j=1..2*n+1) is go(n,x):=sum(a(k,2*n+1)*(x^k)/k!,k=0..infty) = sum(((-1)^(j+1))*exp(j*x),j=1..2*n+1) = exp(x)*(exp((2*n+1)*x)+1)/(exp(x)+1). Via Laplace transformation (see the link under A196837, addendum) one finds the corresponding o.g.f.: Go(n,x) = Po(n,x)/product(1-j*x,j=1..2*n+1) with the numerator polynomial Po(n,x)=sum(a(n,m)*x^m,m=0..2*n). LINKS FORMULA a(n,m)= [x^m](Go(n,x)*product(1-j*x,j=1..2*n+1)), with the o.g.f. Go(n,x) of the sequence a(k,2*n+1):=sum(((-1)^(j+1))*j^k,j=1..2*n+1). See a comment above. a(n,0) = 1, n>=0, and a(n,m) = (-1)^m*(sum(S_{2*i-1,2*i}(2*(n-1),m),i=1..n) + |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). EXAMPLE n\m 0   1   2     3     4       5      6       7       8 0:  1 1:  1  -4   5 2:  1 -12  55  -114    94 3:  1 -24 238 -1248  3661   -5736   3828 4:  1 -40 690 -6700 40053 -151060 351800 -465000, 270576 ... 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(1-j*x,j=1..5). 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. CROSSREFS Cf. A196847, A196837. Sequence in context: A201411 A206282 A082051 * A266699 A234937 A210590 Adjacent sequences:  A196845 A196846 A196847 * A196849 A196850 A196851 KEYWORD sign,easy,tabf AUTHOR Wolfdieter Lang, Oct 27 2011 STATUS approved

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Last modified May 18 19:58 EDT 2021. Contains 344002 sequences. (Running on oeis4.)