A191412 G.f. satisfies: A(x) = x*exp( Sum_{n>=1} A(x^n/(1-x^n))/n ).

1, 1, 3, 9, 34, 139, 643, 3198, 17186, 98438, 598551, 3842893, 25957607, 183858842, 1361853786, 10523285935, 84648613474, 707461189513, 6132612218266, 55050390426042, 510994289786018, 4898133459331104, 48424584171850411

Offset

1,3

Comments

Compare the g.f. to the g.f. P(x) of the partition numbers (A000041):

* P(x) = exp( Sum_{n>=1} (x^n/(1-x^n))/n ),

and to the g.f. R(x) of rooted trees with n nodes (A000081):

* R(x) = x*exp( Sum_{n>=1} R(x^n)/n ).

Consider also the trivial identity:

* B(x) = x*exp( Sum_{n>=1} B(x^n/(1+x^n))/n ) where B(x) = x/(1-x).

Links

Table of n, a(n) for n=1..23.

Example

G.f.: A(x) = x + x^2 + 3*x^3 + 9*x^4 + 34*x^5 + 139*x^6 + 643*x^7 +...
where the g.f. A(x) satisfies:
log(A(x)/x) = A(x/(1-x)) + A(x^2/(1-x^2))/2 + A(x^3/(1-x^3))/3 + A(x^4/(1-x^4))/4 +...
Explicitly, the logarithmic series begins:
log(A(x)/x) = x + 5*x^2/2 + 19*x^3/3 + 93*x^4/4 + 466*x^5/5 + 2633*x^6/6 + 15534*x^7/7 + 97645*x^8/8 +...

Prog

(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x*exp(sum(m=1, n, subst(A, x, x^m/(1-x^m+x*O(x^n)))/m))); polcoeff(A, n)}

Crossrefs

Sequence in context: A353944 A245893 A085686 * A371542 A349017 A246013

Adjacent sequences: A191409 A191410 A191411 * A191413 A191414 A191415

Keyword

nonn

Author

Paul D. Hanna, Jun 01 2011

Status

approved