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.

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

Offset: 0

Author

Wolfdieter Lang, Oct 27 2011

Keywords

• sign
• easy
• tabf

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_{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).

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.

Examples

			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_{j=1..5} (1-j*x).
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.

Formula

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.

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).