A200537 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} A200536(n,k)^2 * x^k] / A(x)^n * x^n/n ), where A200536(n,k) is the coefficient of x^k in (1+3*x+2*x^2)^n.
1, 1, 9, 13, 40, 72, 144, 252, 432, 720, 1152, 1872, 2880, 4608, 6912, 10944, 16128, 25344, 36864, 57600, 82944, 129024, 184320, 285696, 405504, 626688, 884736, 1363968, 1916928, 2949120, 4128768, 6340608, 8847360, 13565952, 18874368, 28901376, 40108032, 61341696, 84934656, 129761280
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 9*x^2 + 13*x^3 + 40*x^4 + 72*x^5 + 144*x^6 +... The logarithm of the g.f. A(x) equals the series: log(A(x)) = (1 + 3^2*x + 2^2*x^2)/A(x) * x + (1 + 6^2*x + 13^2*x^2 + 12^2*x^3 + 4^2*x^4)/A(x)^2 * x^2/2 + (1 + 9^2*x + 62^2*x^2 + 63^2*x^3 + 66^2*x^4 + 36^2*x^5 + 8^2*x^6)/A(x)^3 * x^3/3 + (1 + 12^2*x + 62^2*x^2 + 180^2*x^3 + 321^2*x^4 + 360^2*x^5 + 248^2*x^6 + 96^2*x^7 + 16^2*x^8)/A(x)^4 * x^4/4 + (1 + 15^2*x + 100^2*x^2 + 390^2*x^3 + 985^2*x^4 + 1683^2*x^5 + 1970^2*x^6 + 1560^2*x^7 + 800^2*x^8 + 240^2*x^9 + 32^2*x^10)/A(x)^5 * x^5/5 +... which involves the squares of coefficients A200536(n,k) in (1+3*x+2*x^2)^n.
Programs
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Mathematica
CoefficientList[Series[(1+x)*(1+x^2)*(1+4*x^2)*(1+4*x^3) / (1-2*x^2)^2, {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 11 2015 *) Flatten[{1,1,9,13,40,Table[FullSimplify[9 * 2^(n/2-4) * (-8 - 7*Sqrt[2] + 4*n + 3*Sqrt[2]*n + (-1)^(n+1)*(8 - 7*Sqrt[2] + (-4 + 3*Sqrt[2])*n))],{n,5,40}]}] (* Vaclav Kotesovec, Feb 11 2015 *) -
PARI
{a(n)=polcoeff((1+x)*(1+x^2)*(1+4*x^2)*(1+4*x^3) / ((1-2*x^2)^2+x*O(x^n)),n)} -
PARI
{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1, n, sum(k=0, n, polcoeff((1+3*x+2*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)}
Formula
G.f.: (1+x)*(1+x^2)*(1+4*x^2)*(1+4*x^3) / (1-2*x^2)^2.