A110448 G.f.: A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), where A056045(n) = Sum_{d|n} binomial(n,d).

1, 1, 2, 3, 6, 8, 18, 23, 49, 73, 145, 194, 474, 611, 1331, 2027, 4393, 5919, 14736, 19415, 46487, 68504, 156618, 212055, 560380, 739165, 1833012, 2657837, 6513367, 8743208, 23649777, 31140300, 81276046, 114962333, 293600318, 391926154

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..3337

Formula

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

a(n) ~ c * 2^n / n^(3/2), where c = 2.8176325363130737043447... if n is even and c = 1.784372019603712867208... if n is odd. - Vaclav Kotesovec, Jan 15 2019

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 18*x^6 +...
where A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), or
A(x) = exp(x + 3/2*x^2 + 4/3*x^3 + 11/4*x^4 + 6/5*x^5 +...).
The g.f. can also be expressed as the product:
A(x) = 1/(1-x)*G000108(x^2)*G001764(x^3)*G002293(x^4)*G002294(x^5)*...
where the functions are g.f.s of well-known sequences:
G000108(x) = 1 + x*G000108(x)^2 = g.f. of A000108 ;
G001764(x) = 1 + x*G001764(x)^3 = g.f. of A001764 ;
G002293(x) = 1 + x*G002293(x)^4 = g.f. of A002293 ;
G002294(x) = 1 + x*G002294(x)^5 = g.f. of A002294 ; etc.

Prog

(PARI) {a(n)=polcoeff(exp(x*Ser(vector(n, m, sumdiv(m, d, binomial(m, d))/m))+x*O(x^n)), n)}
(PARI) {a(n)=polcoeff(prod(m=1, n, 1/x*serreverse(x/(1+x^m +x*O(x^n)))), n)}

Crossrefs

Cf. A056045, A174461, A206290.

Cf. A000108, A001764, A002293, A002294, A002295.

Sequence in context: A330442 A103065 A281142 * A005508 A022542 A220418

Adjacent sequences: A110445 A110446 A110447 * A110449 A110450 A110451

Keyword

nonn

Author

Paul D. Hanna, Jul 20 2005, Nov 10 2007

Status

approved