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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. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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(((-1)^j)*j^k,j=1..2*n) is ge(n,x):=sum(a(k,2*n)*(x^k)/k!,k=0..infty) = sum(((-1)^j)*exp(j*x),j=1..2*n) = 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(1-j*x,j=1..2*n) with the numerator polynomial Pe(n,x)=sum(a(n,m)*x^m,m=0..2*(n-1)).

LINKS

Table of n, a(n) for n=1..41.

FORMULA

a(n,m)= [x^m](Ge(n,x)*product(1-j*x,j=1..2*n)/(n*x)), with the o.g.f. Ge(n,x) of the sequence a(k,2*n):=sum(((-1)^j)*j^k,j=1..2*n).See a comment above.

a(n,m) = (-1)^m*sum(S_{2*i-1,2*i}(2*(n-1),m),i=1..n)/n, n>=1, with the (i,j)-family of number triangles S_{i,j}(n,k) defined in a comment to A196845.

EXAMPLE

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(1-j*x,j=1..4).

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.

Sequence in context: A145737 A108763 A061415 * A087455 A294403 A192040

Adjacent sequences:  A196844 A196845 A196846 * A196848 A196849 A196850

KEYWORD

sign,easy,tabf

AUTHOR

Wolfdieter Lang, Oct 27 2011

STATUS

approved

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Last modified August 3 17:21 EDT 2021. Contains 346439 sequences. (Running on oeis4.)