A177762 Beta polynomials (coefficients in descending order, triangle read by rows).

1, 1, 1, -1, 1, -2, -2, 1, -3, -3, 5, 1, -4, -4, 16, 16, 1, -5, -5, 35, 35, -61, 1, -6, -6, 64, 64, -272, -272, 1, -7, -7, 105, 105, -791, -791, 1385, 1, -8, -8, 160, 160, -1856, -1856, 7936, 7936, 1, -9, -9, 231, 231, -3801, -3801, 28839, 28839, -50521

Offset: 0

Author

Peter Luschny, May 13 2010

Keywords

• easy
• sign
• tabf

Comments

beta_n(x) = sum_{k=0..n-1} C(n,k)b(-k)(z-1)^(n-k-1) for n > 0 and beta_0(x)=1. Here b(s) = 2*4^(-s)(zeta(s,1/4)-zeta(s,3/4)) where zeta(s,t) denotes the Hurwitz zeta function.

beta_n(0) are the signed Euler numbers 1,1,-1,-2,5,16,-61,... The sign pattern is the same as in the egf. tanh + sech.

Examples

			1
1
z - 1
z^2 - 2 z - 2
z^3 - 3 z^2 - 3 z + 5
z^4 - 4 z^3 - 4 z^2 + 16 z + 16
z^5 - 5 z^4 - 5 z^3 + 35 z^2 + 35 z - 61
		

Links

• Peter Luschny, Swiss-Knife polynomials and Euler numbers, Blog on OEIS
• Eric Weisstein's World of Mathematics, Dirichlet Beta Function

Crossrefs

Cf. A000111.

Programs

Maple

beta := proc(n, z) option remember; local k;
if n = 0 then 1 else add(`if`(k mod 2 = 1, 0,
binomial(n,k)*beta(k,0)*(z-1)^(n-k-1)),k=0..n-1) fi end:

Mathematica

beta[n_, z_] := beta[n, z] = If[n == 0, 1, Sum[If[OddQ[k], 0, Binomial[n, k]*beta[k, 0]*(z-1)^(n-k-1)], {k, 0, n-1}]];
Table[CoefficientList[beta[n, z], z] // Reverse, {n, 0, 10}] (* Jean-François Alcover, Jun 17 2019, from Maple *)