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_{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.
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.
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_{j=1..5} (1-j*x).
CROSSREFS
KEYWORD
sign,easy,tabf
AUTHOR
Wolfdieter Lang, Oct 27 2011
STATUS
approved