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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.

Original entry on oeis.org

1, 1, -5, 7, 1, -14, 73, -168, 148, 1, -27, 298, -1719, 5473, -9162, 6396, 1, -44, 830, -8756, 56453, -227744, 562060, -778800, 468576, 1, -65, 1865, -31070, 332463, -2385305, 11612795, -37875240, 79269676, -96420480, 52148160, 1, -90, 3647, -87900, 140202
Offset: 1

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Author

Wolfdieter Lang, Oct 27 2011

Keywords

Comments

The row length sequence of this array is A005408(n-1), n >= 1: 1,3,5,7,...
This is the array for the numerator polynomials of the o.g.f. of alternating power sums of the first 2*n positive integers.
The corresponding array for the first 2*n+1 positive integers is found in A196848.
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).
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.

Examples

			n\m 0   1   2     3     4       5      6       7      8
1:  1
2:  1  -5   7
3:  1 -14  73  -168   148
4:  1 -27 298 -1719  5473   -9162   6396
5:  1 -44 830 -8756 56453 -227744 562060 -778800 468576
...
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).
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.

Crossrefs

Cf. A196848, A196837.

Formula

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.
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.

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.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Oct 27 2011

Keywords

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).
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