|
%I #9 Nov 02 2014 12:35:44
%S 1,1,2,15,232,5693,202398,9829771,624964724,50365047225,5016187555114,
%T 604968014349767,86878610741366976,14648881145458377397,
%U 2865572277481996560950,643666405504709227632003,164536267335939429654990988,47489465018413227906492425009
%N G.f.: Sum_{n>=0} n!^2 * x^n / Product_{k=1..n} (1 + k*x)^2.
%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="/A216860/b216860.txt">Table of n, a(n) for n = 0..220</a>
%F a(n) ~ exp(-1) * (n!)^2. - _Vaclav Kotesovec_, Nov 02 2014
%e G.f.: A(x) = 1 + x + 2*x^2 + 15*x^3 + 232*x^4 + 5693*x^5 + 202398*x^6 +...
%e where
%e A(x) = 1 + x/(1+x)^2 + 2!^2*x^2/((1+x)*(1+2*x))^2 + 3!^2*x^3/((1+x)*(1+2*x)*(1+3*x))^2 + 4!^2*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x))^2 +...
%o (PARI) {a(n)=polcoeff(sum(m=0, n, m!^2*x^m/prod(k=1, m, 1+k*x +x*O(x^n))^2), n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A216859.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 17 2012
|
