A177043 Central MacMahon numbers: a(n)=A060187(2*n+1, n+1).

1, 6, 230, 23548, 4675014, 1527092468, 743288515164, 504541774904760, 455522635895576646, 527896878148304296900, 763820398700983273655796, 1349622683586635111555174216, 2859794140516672651686471055900, 7157996663278223282076538528360968

Offset: 0

Author

Roger L. Bagula, May 01 2010

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..201

Crossrefs

Cf. A000108, A060187, A154420.

Programs

Maple

a:= n-> add((-1)^(n-i) *binomial(2*n+1, n-i) *(2*i+1)^(2*n), i=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 05 2011
# With the generating function of the generalized Eulerian polynomials:
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(2*n,2),x,n),n=0..13); # Peter Luschny, May 02 2013

Mathematica

(*A060187*)
p[x_,n_]=(1-x)^(n+1)*Sum[(2*k+1)^n*x^k,{k,0,Infinity}];
f[n_,m_]:=CoefficientList[FullSimplify[ExpandAll[p[x,n]]],x][[m+1]];
a=Table[f[2*n,n],{n,0,20}]

Formula

a(n) ~ sqrt(3) * 2^(4*n+1) * n^(2*n) / exp(2*n). - Vaclav Kotesovec, Sep 30 2014