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%I #11 Mar 04 2014 05:31:05
%S 1,3,7,51,761,17913,688745,56611987,11405877739,4272862207703,
%T 2450039810788461,2224842228379519641,4169966883810355864393,
%U 19139862395982576668262825,166161479603614500915921996017,2206856314384330228779059994929555
%N L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k*(n-k))*x^k = Sum_{n>=1} a(n)*x^n/n.
%C Equals the logarithmic derivative of A207137.
%F a(n) = n*Sum_{k=0..[n/2]} binomial((n-k)^2, k*(n-2*k))/(n-k).
%F Limit n->infinity a(n)^(1/n^2) = ((1-r)^2/(r*(1-2*r)))^((1-3*r)*(1-r)/(3*(1-2*r))) = 1.36198508972775011599..., where r = 0.195220321930105755... is the root of the equation (1-3*r+3*r^2)^(3*(2*r-1)) = (r*(1-2*r))^(4*r-1) * (1-r)^(4*(r-1)). - _Vaclav Kotesovec_, Mar 04 2014
%e L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 51*x^4/4 + 761*x^5/5 + 17913*x^6/6 +...
%e where exponentiation equals the g.f. of A207137:
%e exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 17*x^4 + 171*x^5 + 3171*x^6 +...
%e To illustrate the definition, the l.g.f. equals the series:
%e L(x) = (1 + x)*x + (1 + 4*x + 1*x^2)*x^2/2
%e + (1 + 36*x + 36*x^2 + 1*x^3)*x^3/3
%e + (1 + 560*x + 1820*x^2 + 560*x^3 + 1*x^4)*x^4/4
%e + (1 + 12650*x + 177100*x^2 + 177100*x^3 + 12650*x^4 + 1*x^5)*x^5/5
%e + (1 + 376992*x + 30260340*x^2 + 94143280*x^3 + 30260340*x^4 + 376992*x^5 + 1*x^6)*x^6/6 +...
%t Table[n*Sum[Binomial[(n-k)^2, k*(n-2*k)]/(n-k),{k,0,Floor[n/2]}],{n,1,20}] (* _Vaclav Kotesovec_, Mar 04 2014 *)
%o (PARI) {a(n)=n*polcoeff(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m^2, k*(m-k))*x^k))+x*O(x^n),n)}
%o (PARI) {a(n)=n*sum(k=0,n\2,binomial((n-k)^2,k*(n-2*k))/(n-k))}
%o for(n=1,20,print1(a(n),", "))
%Y Cf. A207137 (exp), A207136, A167006, A228837.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Feb 15 2012
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