A154420 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))).

1, 1, 6, 23, 230, 1682, 23548, 259723, 4675014, 69413294, 1527092468, 28588019814, 743288515164, 16818059163492, 504541774904760, 13397724585164019, 455522635895576646, 13892023109165902550, 527896878148304296900

Offset: 0

Author

Roger L. Bagula, Jan 09 2009

Keywords

• nonn

Comments

Since the center is the maximum in the Pascal, Eulerian and MacMahon triangles, a(n)=MacMahon[n,Floor[n/2]]

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..300
• Peter Luschny, Generalized Eulerian polynomials.

Crossrefs

Cf. A060187, A006551.

Programs

Maple

gf := proc(n, k) local f; f := (x,t) -> x*exp(t*x/k)/(1-x*exp(t*x));
series(f(x,t), t, n+2); ((1-x)/x)^(n+1)*k^n*n!*coeff(%, t, n):
collect(simplify(%), x) end:
seq(coeff(gf(n,1),x,iquo(n,2)),n=0..18); # Middle Eulerian numbers, A006551.
seq(coeff(gf(n,2),x,iquo(n,2)),n=0..18); # Middle midpoint Eulerian numbers.
# Peter Luschny, May 02 2013

Mathematica

p[x_, n_] = 2^n*(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2];
Table[Max[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]], {n, 0, 30}]

Formula

a(n) ~ sqrt(3) * 2^(n+1) * n^n / exp(n). - Vaclav Kotesovec, Oct 28 2021

Extensions

Edited by N. J. A. Sloane, Jan 15 2009