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.
1, 4, 9, 37, 40, 153, 144, 468, 432, 1260, 1152, 3168, 2880, 7632, 6912, 17856, 16128, 40896, 36864, 92160, 82944, 205056, 184320, 451584, 405504, 986112, 884736, 2138112, 1916928, 4608000, 4128768, 9879552, 8847360, 21086208, 18874368, 44826624, 40108032, 94961664, 84934656, 200540160
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 4*x + 9*x^2 + 37*x^3 + 40*x^4 + 153*x^5 + 144*x^6 +... The logarithm of the g.f. A(x) equals the series: log(A(x)) = (2^2 + 3^2*x + x^2)/A(x) * x + (4^2 + 12^2*x + 13^2*x^2 + 6^2*x^3 + x^4)/A(x)^2 * x^2/2 + (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 + (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 + (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 +... which involves the squares of coefficients A200536(n,2*n-k) in (2+3*x+x^2)^n.
Programs
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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)} for(n=0,40,print1(a(n),", ")) -
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)} for(n=0,40,print1(a(n),", "))
Formula
G.f.: (1+4*x)*(1+4*x^2)*(1+x^2)*(1+x^3) / (1-2*x^2)^2.