A195983 - OEIS

%I #20 Nov 19 2014 05:48:10

%S 1,0,1,2,7,32,189,1382,12090,123220,1433760,18750856,272243150,

%T 4344694984,75590125814,1423923579636,28872306351182,626986388773656,

%U 14518030696953650,357068944503961140,9296168815245928682,255409012110904594768,7385107814017356745382

%N G.f.: Product_{n>=1} [Product_{k=0..2*n} (1 - k*x)^(-(-1)^k*binomial(2*n,k))]^(1/n).

%C Compare g.f. to the product:

%C _ exp(x) = Product_{n>=1} [Product_{k=0..n} (1 + k*x)^(-(-1)^k*binomial(n,k))]^(1/n),

%C more explicitly:

%C _ exp(x) = [(1+x)^1/(1+0*x)^1] * [(1+x)^2/((1+0*x)^1*(1+2*x)^1)]^(1/2) * [(1+x)^3*(1+3*x)^1/((1+0*x)^1*(1+2*x)^3)]^(1/3) * [(1+x)^4*(1+3*x)^4/((1+0*x)^1*(1+2*x)^6*(1+4*x)^1)]^(1/4) * ...

%H Vaclav Kotesovec, <a href="/A195983/b195983.txt">Table of n, a(n) for n = 0..320</a>

%F G.f.: A(x) = exp( Sum_{n>=1} 2*A000670(n)*x^(n+1)/(n+1) ) where Sum_{n>=0} A000670(n)*x^n = Sum_{n>=0} n!*x^n/Product_{k=0..n} (1-k*x).

%F a(n) ~ n! / (n^2 * (log(2))^n). - _Vaclav Kotesovec_, Nov 19 2014

%e G.f.: A(x) = 1 + x^2 + 2*x^3 + 7*x^4 + 32*x^5 + 189*x^6 + 1382*x^7 +...

%e where

%e A(x) = [(1-x)^2/((1-0*x)^1*(1-2*x)^1)] * [(1-x)^4*(1-3*x)^4/((1-0*x)^1*(1-2*x)^6*(1-4*x)^1)]^(1/2) * [(1-x)^6*(1-3*x)^20*(1-5*x)^6/((1-0*x)^1*(1-2*x)^15*(1-4*x)^15*(1-6*x)^1)]^(1/3) * ...

%e Also,

%e log(A(x)) = 2*x^2/2 + 6*x^3/3 + 26*x^4/4 + 150*x^5/5 + 1082*x^6/6 + 9366*x^7/7 +...+ 2*A000670(n)*x^(n+1)/(n+1) +...

%e thus, the logarithmic derivative equals the series:

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

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

%o (PARI) /* Using o.g.f. exp( Sum_{n>=1} 2*A000670(n)*x^(n+1)/(n+1) ): */

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

%Y Cf. A000670, A084784.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Sep 25 2011