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%I #13 Jan 09 2024 12:24:24
%S 1,1,6,23,230,1682,23548,259723,4675014,69413294,1527092468,
%T 28588019814,743288515164,16818059163492,504541774904760,
%U 13397724585164019,455522635895576646,13892023109165902550,527896878148304296900
%N Maximal coefficient of MacMahon polynomial (cf. A060187) p(x,n)=2^n*(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2]; that is, a(n) = Max(coefficients(p(x,n))).
%C Since the center is the maximum in the Pascal, Eulerian and MacMahon triangles, a(n)=MacMahon[n,Floor[n/2]]
%H Vaclav Kotesovec, <a href="/A154420/b154420.txt">Table of n, a(n) for n = 0..300</a>
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/EulerianPolynomialsGeneralized">Generalized Eulerian polynomials</a>.
%F a(n) ~ sqrt(3) * 2^(n+1) * n^n / exp(n). - _Vaclav Kotesovec_, Oct 28 2021
%p gf := proc(n, k) local f; f := (x,t) -> x*exp(t*x/k)/(1-x*exp(t*x));
%p series(f(x,t), t, n+2); ((1-x)/x)^(n+1)*k^n*n!*coeff(%, t, n):
%p collect(simplify(%), x) end:
%p seq(coeff(gf(n,1),x,iquo(n,2)),n=0..18); # Middle Eulerian numbers, A006551.
%p seq(coeff(gf(n,2),x,iquo(n,2)),n=0..18); # Middle midpoint Eulerian numbers.
%p # _Peter Luschny_, May 02 2013
%t p[x_, n_] = 2^n*(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2];
%t Table[Max[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]], {n, 0, 30}]
%Y Cf. A060187, A006551.
%K nonn
%O 0,3
%A _Roger L. Bagula_, Jan 09 2009
%E Edited by _N. J. A. Sloane_, Jan 15 2009
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