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

%I A177762 #12 Jun 02 2025 02:55:58
%S A177762 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,
%T A177762 64,64,-272,-272,1,-7,-7,105,105,-791,-791,1385,1,-8,-8,160,160,-1856,
%U A177762 -1856,7936,7936,1,-9,-9,231,231,-3801,-3801,28839,28839,-50521
%N A177762 Beta polynomials (coefficients in descending order, triangle read by rows).
%C A177762 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.
%C A177762 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.
%H A177762 Peter Luschny, Swiss-Knife polynomials and Euler numbers, <a href="http://www.oeis.org/wiki/User:Peter_Luschny/SwissKnifePolynomials">Blog on OEIS</a>
%H A177762 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletBetaFunction.html">Dirichlet Beta Function</a>
%e A177762 1
%e A177762 1
%e A177762 z - 1
%e A177762 z^2 - 2 z - 2
%e A177762 z^3 - 3 z^2 - 3 z + 5
%e A177762 z^4 - 4 z^3 - 4 z^2 + 16 z + 16
%e A177762 z^5 - 5 z^4 - 5 z^3 + 35 z^2 + 35 z - 61
%p A177762 beta := proc(n, z) option remember; local k;
%p A177762 if n = 0 then 1 else add(`if`(k mod 2 = 1, 0,
%p A177762 binomial(n,k)*beta(k,0)*(z-1)^(n-k-1)),k=0..n-1) fi end:
%t A177762 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}]];
%t A177762 Table[CoefficientList[beta[n, z], z] // Reverse, {n, 0, 10}] (* _Jean-François Alcover_, Jun 17 2019, from Maple *)
%Y A177762 Cf. A000111.
%K A177762 easy,sign,tabf
%O A177762 0,6
%A A177762 _Peter Luschny_, May 13 2010