A203266 G.f.: A(x) = exp( Sum_{n>=1} G_n(x^n)^2 * x^n/n ) such that G_n(x^n) = Product_{k=0..n-1} A(u^k*x) where u is an n-th root of unity.

1, 1, 3, 10, 43, 172, 852, 3719, 19290, 90469, 481825, 2295973, 12812880, 62122518, 346770241, 1744884177, 9830723932, 49268101457, 285020577850, 1438021808125, 8300553742452, 42955378420263, 247456097485363, 1271978597532857, 7520668963944205, 38700952324615078

Offset

0,3

Links

Table of n, a(n) for n=0..25.

Formula

G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} 2*A203265(n*k)*x^(n*k)/k ) ) where A(x) = exp( Sum_{n>=1} A203265(n)*x^n/n ).

The logarithmic derivative yields A203265.

Example

G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 43*x^4 + 172*x^5 + 852*x^6 + 3719*x^7 +...
G.f.: A(x) = exp( Sum_{n>=1} A203265(n) * x^n/n ),
where A(x) = exp( Sum_{n>=1} G_n(x^n)^2 * x^n/n )
and G_n(x) = exp( Sum_{k>=1} A203265(n*k)*x^k/k ), which begin:
G_1(x) = A(x);
G_2(x) = 1 + 5*x + 75*x^2 + 1518*x^3 + 34663*x^4 + 867760*x^5 +...;
G_3(x) = 1 + 22*x + 2019*x^2 + 214648*x^3 + 31221037*x^4 +...;
G_4(x) = 1 + 125*x + 59771*x^2 + 40659310*x^3 + 31438395303*x^4 +...;
G_5(x) = 1 + 576*x + 1760688*x^2 + 6380121685*x^3 +...;
G_6(x) = 1 + 3554*x + 57073923*x^2 + 1295238092004*x^3 +...;
G_7(x) = 1 + 16843*x + 1719312892*x^2 + 212162358939394*x^3 +...;
G_8(x) = 1 + 103917*x + 56284535547*x^2 + 44125115136389518*x^3 +...;
...
Also, G_n(x^n) = Product_{k=0..n-1} A(u^k*x) where u = n-th root of unity:
G_2(x^2) = A(x)*A(-x);
G_3(x^3) = A(x)*A(u*x)*A(u^2*x) where u = exp(2*Pi*I/3);
G_4(x^4) = A(x)*A(u*x)*A(u^2*x)*A(u^3*x) where u = exp(2*Pi*I/4);
...
The logarithmic derivative of this sequence yields A203265:
A203265 = [1,5,22,125,576,3554,16843,103917,521338,3189600,...].

Prog

(PARI) {a(n)=local(L=vector(n, i, 1)); for(i=1, n, L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, floor(n/m), 2*L[m*k]*x^(m*k)/k)+x*O(x^n)))))); polcoeff(exp(x*Ser(vector(n, m, L[m]/m))), n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, x^m/m*round(prod(k=0, m-1, subst(A^2, x, exp(2*Pi*I*k/m)*x+x*O(x^n))))))); polcoeff(A, n)}

Crossrefs

Cf. A203265 (log), A203254, A203268.

Sequence in context: A288813 A306237 A042545 * A151084 A151085 A082936

Adjacent sequences: A203263 A203264 A203265 * A203267 A203268 A203269

Keyword

nonn

Author

Paul D. Hanna, Dec 30 2011

Status

approved