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 A177762 Beta polynomials (coefficients in descending order, triangle read by rows) 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 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. LINKS Peter Luschny, Swiss-Knife polynomials and Euler numbers, Blog on OEIS Eric Weisstein's World of Mathematics, Dirichlet Beta Function EXAMPLE 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 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 *) CROSSREFS Cf. A000111. Sequence in context: A355668 A023990 A117894 * A109380 A167754 A011020 Adjacent sequences: A177759 A177760 A177761 * A177763 A177764 A177765 KEYWORD easy,sign,tabf AUTHOR Peter Luschny, May 13 2010 STATUS approved

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Last modified March 29 12:04 EDT 2023. Contains 361599 sequences. (Running on oeis4.)