A203254 G.f.: A(x) = exp( Sum_{n>=1} G_n(x^n)*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, 2, 4, 10, 22, 62, 146, 422, 1084, 3160, 8064, 25190, 65204, 198652, 545790, 1680122, 4495548, 14352768, 38665478, 122530052, 343978146, 1072985932, 2947659006, 9662067644, 26573691092, 84395544446, 241295995524, 769819399580, 2140972333774, 7039688293036, 19579468840840

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} A203253(n*k)*x^(n*k)/k ) ) where A(x) = exp( Sum_{n>=1} A203253(n)*x^n/n ).

The logarithmic derivative yields A203253.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 22*x^5 + 62*x^6 + 146*x^7 +...
G.f.: A(x) = exp( Sum_{n>=1} A203253(n)*x^n/n ),
where A(x) = exp( Sum_{n>=1} G_n(x^n)*x^n/n )
and G_n(x) = exp( Sum_{k>=1} A203253(n*k)*x^k/k ), which begin:
G_1(x) = A(x);
G_2(x) = 1 + 3*x + 16*x^2 + 104*x^3 + 724*x^4 + 5428*x^5 + 44080*x^6 +...;
G_3(x) = 1 + 7*x + 122*x^2 + 2128*x^3 + 52330*x^4 + 1109386*x^5 +...;
G_4(x) = 1 + 23*x + 1080*x^2 + 67944*x^3 + 4595792*x^4 +...;
G_5(x) = 1 + 51*x + 8582*x^2 + 1482524*x^3 + 355949360*x^4 +...;
G_6(x) = 1 + 195*x + 89752*x^2 + 53146664*x^3 + 36695632888*x^4 +...;
G_7(x) = 1 + 435*x + 705756*x^2 + 1208493276*x^3 +...;
G_8(x) = 1 + 1631*x + 7232560*x^2 + 44157620896*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(I*x)*A(I^2*x)*A(I^3*x) where I^2 = -1;
...
The logarithmic derivative of this sequence yields A203253:
A203253 = [1,3,7,23,51,195,435,1631,4165,14563,34761,141479,...].

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

Crossrefs

Cf. A203253 (log), A000081.

Sequence in context: A005962 A208062 A179468 * A076875 A179490 A337520

Adjacent sequences: A203251 A203252 A203253 * A203255 A203256 A203257

Keyword

nonn

Author

Paul D. Hanna, Dec 30 2011

Status

approved