%I #9 May 09 2014 10:31:19
%S 1,1,2,9,55,412,3665,37809,443998,5848921,85425959,1370144160,
%T 23941364521,452710417321,9210564625442,200626664154849,
%U 4658472162245695,114865936425213532,2997499707147860825,82533717939413618649,2391252655460083134718,72723156542550310492081,2316342951911550838935119
%N G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k - x) / (1 - k*x).
%C Compare to a g.f. of the Fibonacci numbers (A000045):
%C Sum_{n>=0} x^n * Product_{k=1..n} (k + x)/(1 + k*x) = 1/(1-x-x^2).
%H Vaclav Kotesovec, <a href="/A231172/b231172.txt">Table of n, a(n) for n = 0..420</a>
%F a(n) = Sum_{k=0..n} A231171(n,k)*(-1)^k for n>=0.
%F Limit n->infinity (a(n)/n!)^(1/n) = 1/log(2). - _Vaclav Kotesovec_, May 09 2014
%e G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 55*x^4 + 412*x^5 + 3665*x^6 +...
%e where
%e A(x) = 1 + x*(1-x)/(1-x) + x^2*(1-x)*(2-x)/((1-x)*(1-2*x)) + x^3*(1-x)*(2-x)*(3-x)/((1-x)*(1-2*x)*(1-3*x)) + x^4*(1-x)*(2-x)*(3-x)*(4-x)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) +...
%o (PARI) {a(n)=polcoeff(sum(m=0, n, x^m*prod(k=1, m, k-x +x*O(x^n))/prod(k=1, m, 1-k*x +x*O(x^n))), n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A231171, A231173.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 05 2013
|