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%I #12 Nov 07 2015 02:20:09
%S 0,1,0,-2,0,40,0,-3024,0,604800,0,-262690560,0,217945728000,0,
%T -315323879270400,0,742997162299392000,0,-2703345607134653644800,0,
%U 14552624755991316234240000,0,-111913707637423660385894400000,0
%N a(n) = 2*(n+2)!*(zeta(-2*n)-zeta(-n)), zeta(n) the Riemann zeta function.
%C Old name was: A polynomial expansion: p(x,t)=-Exp[t]*(-1 + Exp[x])/(-1 + Exp[t]).
%C The expansion is the solution for integer q of: q*Exp[x*t]/(q - 1 + Exp[x]) - Exp[t*(1 + x)] = 0. (See the Mathematica program.) That result is a generalized Euler number in q as a Pascal expansion. For higher Sierpinski-Pascal levels (Eulerian and MacMahon) this results in polynomials.
%C For n >= 9, a(n) is divisible by 2*10^(floor(n/5)-1). - _G. C. Greubel_, Nov 06 2015
%H G. C. Greubel, <a href="/A178235/b178235.txt">Table of n, a(n) for n = 0..200</a>
%p A178235 := n -> 2*(n+2)!*(Zeta(-2*n)-Zeta(-n)); seq(A178235(n),n=0..24); # _Peter Luschny_, Jul 14 2013
%t p[t_] = -Exp[t]*(-1 + Exp[x])/(-1 + Exp[t]); Table[ FullSimplify[ExpandAll[(2*(n + 2)!n!/(1 - Exp[x]))*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], {n, 0, 30}]
%K sign
%O 0,4
%A _Roger L. Bagula_, May 23 2010
%E Edited, new name and a(0) changed to 0 by _Peter Luschny_, Jul 14 2013
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