A216859 - OEIS

%I #9 Nov 01 2014 04:33:11

%S 1,1,1,3,65,1871,69885,3466339,222981385,18102473271,1811907033269,

%T 219290184518315,31573338878091585,5334099790769759551,

%U 1045025926871985755053,235016617680587793977331,60133997212733124023350265,17369999617568255471165082311

%N G.f.: Sum_{n>=0} n!^2 * x^n / Product_{k=1..2*n} (1 + k*x).

%C Compare g.f. to: 1/(1-x) = Sum_{n>=0} n!*x^n/Product_{k=1..n} (1 + k*x).

%H Vaclav Kotesovec, <a href="/A216859/b216859.txt">Table of n, a(n) for n = 0..220</a>

%F a(n) ~ exp(-2) * (n!)^2. - _Vaclav Kotesovec_, Nov 01 2014

%e G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 65*x^4 + 1871*x^5 + 69885*x^6 +...

%e where

%e A(x) = 1 + x/((1+x)*(1+2*x)) + 2!^2*x^2/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) +

%e 3!^2*x^3/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)*(1+5*x)*(1+6*x)) +

%e 4!^2*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)*(1+5*x)*(1+6*x)*(1+7*x)*(1+8*x)) +...

%o (PARI) {a(n)=polcoeff(sum(m=0, n, m!^2*x^m/prod(k=1, 2*m, 1+k*x +x*O(x^n)) ), n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A208885, A216860.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Sep 17 2012