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A206851 L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2)*x^k = Sum_{n>=1} a(n)*x^n/n.

%I A206851 #21 Jan 27 2019 07:31:36
%S A206851 1,3,7,15,231,2763,37773,3347359,145164760,15115517783,5300285945494,
%T A206851 841490209145991,700215432847179640,821522962294608211319,
%U A206851 580955012898669141073842,3240132942509582109732641935,12114306457535986210506222037102
%N A206851 L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2)*x^k = Sum_{n>=1} a(n)*x^n/n.
%C A206851 Equals the logarithmic derivative of A206850.
%H A206851 Seiichi Manyama, <a href="/A206851/b206851.txt">Table of n, a(n) for n = 1..94</a>
%F A206851 a(n) = n * Sum_{k=0..[n/2]} binomial((n-k)^2, k^2)/(n-k).
%F A206851 Limit n->infinity a(n)^(1/n^2) = (1-2*r)^r / r^(2*r) = 1.2915356633069917227119166349..., where r = A323778 = 0.365498498219858044579... is the root of the equation (1-r)^(2-2*r) * r^(2*r) = 1-2*r. - _Vaclav Kotesovec_, Mar 03 2014
%e A206851 L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 231*x^5/5 + 2763*x^6/6 +...
%e A206851 where exponentiation yields the g.f. of A206850:
%e A206851 exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 56*x^5 + 522*x^6 + 5972*x^7 +...
%e A206851 By definition, the l.g.f. equals the series:
%e A206851 L(x) = (C(1,0) + C(1,1)*x)*x
%e A206851 + (C(4,0) + C(4,1)*x + C(4,4)*x^2)*x^2/2
%e A206851 + (C(9,0) + C(9,1)*x + C(9,4)*x^2 + C(9,9)*x^3)*x^3/3
%e A206851 + (C(16,0) + C(16,1)*x + C(16,4)*x^2 + C(16,9)*x^3 + C(16,16)*x^4)*x^4/4
%e A206851 + (C(25,0) + C(25,1)*x + C(25,4)*x^2 + C(25,9)*x^3 + C(25,16)*x^4 + C(25,25)*x^5)*x^5/5 +...
%e A206851 More explicitly,
%e A206851 L(x) = (1 + 1*x)*x + (1 + 4*x + 1*x^2)*x^2/2
%e A206851 + (1 + 9*x + 126*x^2 + 1*x^3)*x^3/3
%e A206851 + (1 + 16*x + 1820*x^2 + 11440*x^3 + 1*x^4)*x^4/4
%e A206851 + (1 + 25*x + 12650*x^2 + 2042975*x^3 + 2042975*x^4 + 1*x^5)*x^5/5
%e A206851 + (1 + 36*x + 58905*x^2 + 94143280*x^3 + 7307872110*x^4 + 600805296*x^5 + 1*x^6)*x^6/6 +...
%t A206851 Table[n*Sum[Binomial[(n-k)^2, k^2]/(n-k),{k,0,Floor[n/2]}], {n, 1, 20}] (* _Vaclav Kotesovec_, Mar 03 2014 *)
%o A206851 (PARI) {a(n)=n*polcoeff(sum(m=1,n,x^m/m*sum(k=0,m,binomial(m^2,k^2)*x^k)+x*O(x^n)),n)}
%o A206851 (PARI) {a(n)=n*sum(k=0,n\2, binomial((n-k)^2, k^2)/(n-k))}
%o A206851 for(n=1, 20, print1(a(n), ", "))
%Y A206851 Cf. A206850 (exp), A206847, A206849, A123165.
%K A206851 nonn
%O A206851 1,2
%A A206851 _Paul D. Hanna_, Feb 15 2012