A203268 G.f.: A(x) = exp( Sum_{n>=1} G_n(x^n)^3 * 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, 4, 19, 116, 683, 4818, 31126, 232058, 1598611, 12315375, 86887285, 695017086, 4999457900, 40344295044, 298468091712, 2434392979661, 18077507384936, 150454415661096, 1125745880242406, 9386869540033292, 71518155964958242, 597727034006054509

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} 3*A203267(n*k)*x^(n*k)/k ) ) where A(x) = exp( Sum_{n>=1} A203267(n)*x^n/n ).

The logarithmic derivative yields A203267.

Example

G.f.: A(x) = 1 + x + 4*x^2 + 19*x^3 + 116*x^4 + 683*x^5 + 4818*x^6 +...
G.f.: A(x) = exp( Sum_{n>=1} A203267(n) * x^n/n ),
where A(x) = exp( Sum_{n>=1} G_n(x^n)^3 * x^n/n )
and G_n(x) = exp( Sum_{k>=1} A203267(n*k)*x^k/k ), which begin:
G_1(x) = A(x);
G_2(x) = 1 + 7*x + 210*x^2 + 8837*x^3 + 427910*x^4 + 22758491*x^5 +...;
G_3(x) = 1 + 46*x + 12280*x^2 + 4087909*x^3 + 1805475734*x^4 +...;
G_4(x) = 1 + 371*x + 776202*x^2 + 2360146453*x^3 +...;
G_5(x) = 1 + 2611*x + 49859649*x^2 + 1211412677799*x^3 +...;
G_6(x) = 1 + 22444*x + 3385662240*x^2 + 742868246890817*x^3 +...;
G_7(x) = 1 + 163010*x + 223920974239*x^2 + 396998122840515180*x^3 +...;
G_8(x) = 1 + 1414763*x + 15479260324770*x^2 + 249608398400792533605*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 A203267:
A203267 = [1,7,46,371,2611,22444,163010,1414763,10666423,...].

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), 3*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^3, x, exp(2*Pi*I*k/m)*x+x*O(x^n))))))); polcoeff(A, n)}

Crossrefs

Cf. A203267 (log), A203254, A203266.

Sequence in context: A013185 A353607 A186359 * A209673 A261497 A217989

Adjacent sequences: A203265 A203266 A203267 * A203269 A203270 A203271

Keyword

nonn

Author

Paul D. Hanna, Dec 30 2011

Status

approved