A251689 - OEIS

%I #5 Feb 25 2015 22:24:24

%S 1,4,9,37,40,153,144,468,432,1260,1152,3168,2880,7632,6912,17856,

%T 16128,40896,36864,92160,82944,205056,184320,451584,405504,986112,

%U 884736,2138112,1916928,4608000,4128768,9879552,8847360,21086208,18874368,44826624,40108032,94961664,84934656,200540160

%N 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 G.f.: (1+4*x)*(1+4*x^2)*(1+x^2)*(1+x^3) / (1-2*x^2)^2.

%e G.f.: A(x) = 1 + 4*x + 9*x^2 + 37*x^3 + 40*x^4 + 153*x^5 + 144*x^6 +...

%e The logarithm of the g.f. A(x) equals the series:

%e log(A(x)) = (2^2 + 3^2*x + x^2)/A(x) * x +

%e (4^2 + 12^2*x + 13^2*x^2 + 6^2*x^3 + x^4)/A(x)^2 * x^2/2 +

%e (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 (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 (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 which involves the squares of coefficients A200536(n,2*n-k) in (2+3*x+x^2)^n.

%o (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 for(n=0,40,print1(a(n),", "))

%o (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 for(n=0,40,print1(a(n),", "))

%Y Cf. A200537, A200536, A251688.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 23 2015