A126152 - OEIS

%I #13 May 30 2015 07:42:43

%S 1,4,36,744,28536,1736064,152914176,18372559104,2885671339776,

%T 573765893121024,140835811776316416,41820352964911908864,

%U 14774712204104658671616,6124078747943873540112384

%N Main diagonal of symmetric triangle A126150: a(n) = A126150(n,n).

%H Vaclav Kotesovec, <a href="/A126152/b126152.txt">Table of n, a(n) for n = 0..232</a>

%F a(n) = Sum_{k, 0<=k<=n} A130847(n,k)*3^k. - _Philippe Deléham_, Jul 22 2007

%F G.f.: 1/(1 - 4*x/(1-5*x/(1 - 21*x/(1-22*x/(1 - 50*x/(1-51*x/(1 - 91*x/(1-92*x/(1 -...)))))))))))), a continued fraction involving even-indexed pentagonal numbers A000326. - _Paul D. Hanna_, Feb 15 2012

%F a(n) ~ Gamma(1/3) * 2^(3*n+7/3) * 3^(n+3/2) * n^(2*n+7/6) / (exp(2*n) * Pi^(2*n+13/6)). - _Vaclav Kotesovec_, May 30 2015

%o (PARI) /* Continued fraction involving even-indexed pentagonal numbers: */

%o {a(n)=local(CF=1+x*O(x),m,P); for(k=1, n,m=2*((n-k)\2+1);P=m*(3*m-1)/2-((n-k+1)%2); CF=1/(1-P*x*CF)); polcoeff(CF, n, x)}

%o for(n=0,20,print1(a(n),","))

%Y Cf. A126150; A126151 (column 0), A126153 (diagonal).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 19 2006