A134774 G.f.: A(x) = Product_{n>=1} G(x^n,n)^n where G(x,n) = 1 + x*G(x,n)^n.

1, 1, 3, 6, 15, 26, 66, 110, 253, 460, 966, 1680, 3732, 6304, 13073, 23539, 47548, 82362, 171463, 293578, 597934, 1056830, 2105424, 3654919, 7533609, 12915780, 26112978, 46033557, 92504870, 160298673, 330468463, 568239653, 1161488784

Offset

0,3

Links

Table of n, a(n) for n=0..32.

Formula

G.f.: A(x) = exp( Sum_{n>=1} A105862(n)/n*x^n ), where A105862(n) = Sum_{d|n} binomial(n,d)*n/gcd(n,d).

G.f.: A(x) = Product_{n>=1} [ Series_Reversion( x/(1 + x^n) )/x ]^n.

Example

G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 26*x^5 + 66*x^6 +...
G.f.: A(x) = 1/(1-x) * G(x^2,2)^2 * G(x^3,3)^3 * G(x^4,4)^4 *...
where the functions G(x,n) are g.f.s of well-known sequences:
G(x,2) = g.f. of A000108 = 1 + x*G(x,2)^2;
G(x,3) = g.f. of A001764 = 1 + x*G(x,3)^3;
G(x,4) = g.f. of A002293 = 1 + x*G(x,4)^4 ; etc.
Explicitly, the product yielding the g.f. A(x) begins:
A(x) = [1 + x + x^2 + x^3 +...] * [1 + 2*x^2 + 5*x^4 + 14*x^6 +...] * [1 + 3*x^3 + 12*x^6 + 55*x^9 +...] * [1 + 4*x^4 + 22*x^8 + 140*x^12 +...] * ...

Prog

(PARI) a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, x^m*sumdiv(m, d, binomial(m, d)/gcd(m, d)))), n))
(PARI) a(n)=polcoeff(prod(m=1, n, (1/x*serreverse(x/(1+x^m +x*O(x^n))))^m), n)

Crossrefs

Cf. A105862 (log(A(x)); A056045 (variant); A000108 (Catalan), A001764, A002293.

Sequence in context: A282123 A281420 A280839 * A264686 A337665 A165729

Adjacent sequences: A134771 A134772 A134773 * A134775 A134776 A134777

Keyword

nonn

Author

Paul D. Hanna, Nov 11 2007

Status

approved