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A251689 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} A200536(n,2*n-k)^2 * x^k] / A(x)^n * x^n/n ), where A200536(n,2*n-k) is the coefficient of x^k in (2+3*x+x^2)^n.

%I A251689 #5 Feb 25 2015 22:24:24
%S A251689 1,4,9,37,40,153,144,468,432,1260,1152,3168,2880,7632,6912,17856,
%T A251689 16128,40896,36864,92160,82944,205056,184320,451584,405504,986112,
%U A251689 884736,2138112,1916928,4608000,4128768,9879552,8847360,21086208,18874368,44826624,40108032,94961664,84934656,200540160
%N A251689 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} A200536(n,2*n-k)^2 * x^k] / A(x)^n * x^n/n ), where A200536(n,2*n-k) is the coefficient of x^k in (2+3*x+x^2)^n.
%F A251689 G.f.: (1+4*x)*(1+4*x^2)*(1+x^2)*(1+x^3) / (1-2*x^2)^2.
%e A251689 G.f.: A(x) = 1 + 4*x + 9*x^2 + 37*x^3 + 40*x^4 + 153*x^5 + 144*x^6 +...
%e A251689 The logarithm of the g.f. A(x) equals the series:
%e A251689 log(A(x)) = (2^2 + 3^2*x + x^2)/A(x) * x +
%e A251689 (4^2 + 12^2*x + 13^2*x^2 + 6^2*x^3 + x^4)/A(x)^2 * x^2/2 +
%e A251689 (8^2 + 36^2*x + 66^2*x^2 + 63^2*x^3 + 33^2*x^4 + 9^2*x^5 + x^6)/A(x)^3 * x^3/3 +
%e A251689 (16^2 + 96^2*x + 248^2*x^2 + 360^2*x^3 + 321^2*x^4 + 180^2*x^5 + 62^2*x^6 + 12^2*x^7 + x^8)/A(x)^4 * x^4/4 +
%e A251689 (32^2 + 240^2*x + 800^2*x^2 + 1560^2*x^3 + 1970^2*x^4 + 1683^2*x^5 + 985^2*x^6 + 390^2*x^7 + 100^2*x^8 + 15^2*x^9 + x^10)/A(x)^5 * x^5/5 +...
%e A251689 which involves the squares of coefficients A200536(n,2*n-k) in (2+3*x+x^2)^n.
%o A251689 (PARI) {a(n)=polcoeff( (1+4*x)*(1+4*x^2)*(1+x^2)*(1+x^3) / ((1-2*x^2)^2 +x*O(x^n)), n)}
%o A251689 for(n=0,40,print1(a(n),", "))
%o A251689 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, n, polcoeff((2+3*x+x^2+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 A251689 for(n=0,40,print1(a(n),", "))
%Y A251689 Cf. A200537, A200536, A251688.
%K A251689 nonn
%O A251689 0,2
%A A251689 _Paul D. Hanna_, Feb 23 2015