OFFSET
0,3
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.
FORMULA
a(n) ~ sqrt(3) * 2^(n+1) * n^n / exp(n). - Vaclav Kotesovec, Oct 28 2021
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Roger L. Bagula, Jan 09 2009
EXTENSIONS
Edited by N. J. A. Sloane, Jan 15 2009
STATUS
approved