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A251688 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1+2*x)^n*(1+3*x)^n.

%I A251688 #9 Mar 03 2015 03:31:41
%S A251688 1,1,25,61,336,1200,3600,13500,32400,118800,259200,939600,1944000,
%T A251688 6998400,13996800,50155200,97977600,349920000,671846400,2393452800,
%U A251688 4534963200,16124313600,30233088000,107327462400,199538380800,707454259200,1306069401600,4625662464000,8489451110400
%N A251688 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1+2*x)^n*(1+3*x)^n.
%C A251688 More generally, if G(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / G(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1 + a*x)^n*(1 + b*x)^n, then G(x) = (1+x)*(1 + a^2*x^2)*(1 + b^2*x^2)*(1 + a^2*b^2*x^3) / (1 - a*b*x^2)^2; here a=2, b=3.
%C A251688 More generally, if G(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / G(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (p + q*x)^n*(r + s*x)^n, then G(x) = (1 + p^2*r^2*x)*(1 + p^2*s^2*x^2)*(1 + q^2*r^2*x^2)*(1 + q^2*s^2*x^3) / (1 - p*q*r*s*x^2)^2.
%F A251688 G.f.: (1+x)*(1+4*x^2)*(1+9*x^2)*(1+36*x^3) / (1-6*x^2)^2.
%e A251688 G.f.: A(x) = 1 + x + 25*x^2 + 61*x^3 + 336*x^4 + 1200*x^5 + 3600*x^6 +...
%e A251688 where
%e A251688 log(A(x)) = (1 + 5^2*x + 6^2*x^2)/A(x) * x +
%e A251688 (1 + 10^2*x + 37^2*x^2 + 60^2*x^3 + 36^2*x^4)/A(x)^2 * x^2/2 +
%e A251688 (1 + 15^2*x + 93^2*x^2 + 305^2*x^3 + 558^2*x^4 + 540^2*x^5 + 216^2*x^6)/A(x)^3 * x^3/3 +
%e A251688 (1 + 20^2*x + 174^2*x^2 + 860^2*x^3 + 2641^2*x^4 + 5160^2*x^5 + 6264^2*x^6 + 4320^2*x^7 + 1296^2*x^8)/A(x)^4 * x^4/4 +
%e A251688 (1 + 25^2*x + 280^2*x^2 + 1850^2*x^3 + 7985^2*x^4 + 23525^2*x^5 + 47910^2*x^6 + 66600^2*x^7 + 60480^2*x^8 + 32400^2*x^9 + 7776^2*x^10)/A(x)^5 * x^5/5 +...
%e A251688 which involves the squares of coefficients in (1 + 5*x + 6*x^2)^n.
%o A251688 (PARI) {a(n)=polcoeff( (1+x)*(1+4*x^2)*(1+9*x^2)*(1+36*x^3) / ((1-6*x^2)^2 +x*O(x^n)), n)}
%o A251688 for(n=0, 40, print1(a(n), ", "))
%o A251688 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, n, polcoeff(((1+2*x)*(1+3*x) +x*O(x^k))^m, k)^2 *x^k) *x^m/(A+x*O(x^n))^m/m)+x*O(x^n))); polcoeff(A, n)}
%o A251688 for(n=0, 40, print1(a(n), ", "))
%Y A251688 Cf. A251689, A200537, A251687.
%K A251688 nonn
%O A251688 0,3
%A A251688 _Paul D. Hanna_, Feb 25 2015