A141424 Numerators of second column of the inverse of the triangle of polynomial coefficients P(0,x)=1, 2P(n,x)=(1+x)*[(1+x)^(n-1)+x^(n-1)].

1, -3, 3, -5, 5, -7, 7, 3, -3, -121, 121, 1261, -1261, -20583, 20583, 888403, -888403, -24729925, 24729925, 862992399, -862992399, -36913939769, 36913939769, 1899853421885, -1899853421885, -115841483491323, 115841483491323, 8258802033519361

Offset: 0

Author

Paul Curtz, Aug 06 2008

Keywords

• sign

Comments

For the denominators see A053644.

The P(n,x) polynomials are based on the Euler polynomials and the inverse matrix of their coefficients is described in Example section of A133135. First column is A033999, third column is A133135.

Crossrefs

Cf. A051717.

Programs

Mathematica

max = 27; p[0] = 1; p[n_] := (1+x)*((1+x)^(n-1)+x^(n-1))/2; t = Table[Coefficient[p[n], x, k], {n, 0, max+2}, {k, 0, max+2}]; a[n_] := Inverse[t][[All, 2]][[n+2]] // Numerator; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Dec 16 2013 *)

PARI

lista(n) = {m = matrix(n, n); m[1, 1] = 1; for (i=2, n, pol = (1+x)*((1+x)^(i-2)+x^(i-2))/2; for (j=1, n, m[i, j] = polcoeff(pol, j-1, x););); m = 1/m; for (i=2, n, print1(numerator(m[i, 2]), ", ");); print();} \\ Michel Marcus, Aug 16 2013

Extensions

Edited by Michel Marcus, Aug 16 2013