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

%I A177043 #20 Apr 03 2017 02:56:42
%S A177043 1,6,230,23548,4675014,1527092468,743288515164,504541774904760,
%T A177043 455522635895576646,527896878148304296900,763820398700983273655796,
%U A177043 1349622683586635111555174216,2859794140516672651686471055900,7157996663278223282076538528360968
%N A177043 Central MacMahon numbers: a(n)=A060187(2*n+1, n+1).
%H A177043 Robert Israel, <a href="/A177043/b177043.txt">Table of n, a(n) for n = 0..201</a>
%F A177043 a(n) ~ sqrt(3) * 2^(4*n+1) * n^(2*n) / exp(2*n). - _Vaclav Kotesovec_, Sep 30 2014
%p A177043 a:= n-> add((-1)^(n-i) *binomial(2*n+1, n-i) *(2*i+1)^(2*n), i=0..n):
%p A177043 seq(a(n), n=0..20);  # _Alois P. Heinz_, Dec 05 2011
%p A177043 # With the generating function of the generalized Eulerian polynomials:
%p A177043 gf := proc(n, k) local f; f := (x,t) -> x*exp(t*x/k)/(1-x*exp(t*x));
%p A177043 series(f(x,t), t, n+2); ((1-x)/x)^(n+1)*k^n*n!*coeff(%, t, n):
%p A177043 collect(simplify(%),x) end: seq(coeff(gf(2*n,2),x,n),n=0..13); # _Peter Luschny_, May 02 2013
%t A177043 (*A060187*)
%t A177043 p[x_,n_]=(1-x)^(n+1)*Sum[(2*k+1)^n*x^k,{k,0,Infinity}];
%t A177043 f[n_,m_]:=CoefficientList[FullSimplify[ExpandAll[p[x,n]]],x][[m+1]];
%t A177043 a=Table[f[2*n,n],{n,0,20}]
%Y A177043 Cf. A000108, A060187, A154420.
%K A177043 nonn
%O A177043 0,2
%A A177043 _Roger L. Bagula_, May 01 2010