A177043 - OEIS

%I #20 Apr 03 2017 02:56:42

%S 1,6,230,23548,4675014,1527092468,743288515164,504541774904760,

%T 455522635895576646,527896878148304296900,763820398700983273655796,

%U 1349622683586635111555174216,2859794140516672651686471055900,7157996663278223282076538528360968

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

%H Robert Israel, <a href="/A177043/b177043.txt">Table of n, a(n) for n = 0..201</a>

%F a(n) ~ sqrt(3) * 2^(4*n+1) * n^(2*n) / exp(2*n). - _Vaclav Kotesovec_, Sep 30 2014

%p a:= n-> add((-1)^(n-i) *binomial(2*n+1, n-i) *(2*i+1)^(2*n), i=0..n):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Dec 05 2011

%p # With the generating function of the generalized Eulerian polynomials:

%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: seq(coeff(gf(2*n,2),x,n),n=0..13); # _Peter Luschny_, May 02 2013

%t (*A060187*)

%t p[x_,n_]=(1-x)^(n+1)*Sum[(2*k+1)^n*x^k,{k,0,Infinity}];

%t f[n_,m_]:=CoefficientList[FullSimplify[ExpandAll[p[x,n]]],x][[m+1]];

%t a=Table[f[2*n,n],{n,0,20}]

%Y Cf. A000108, A060187, A154420.

%K nonn

%O 0,2

%A _Roger L. Bagula_, May 01 2010