A161630 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)) ).

1, 1, 3, 19, 181, 2321, 37501, 731935, 16758393, 440525377, 13077834841, 432796650551, 15799794395749, 630773263606513, 27339525297079269, 1278550150117141231, 64171287394646697841, 3440711053857464325377

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..373

Vaclav Kotesovec, Asymptotic of sequences A161630, A212722, A212917 and A245265

Formula

a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(n-1,n-k).

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then

a(n,m) = Sum_{k=0..n} n! * m*(n-k+m)^(k-1)/k! * C(n-1,n-k).

E.g.f. satisfies: A(x) = exp(x) * A(x)^(x*A(x)). - Paul D. Hanna, Aug 02 2013

a(n) ~ n^(n-1) * (1+2*c)^(n+1/2) / (sqrt(1+c) * 2^(2*n+2) * exp(n) * c^(2*n+3/2)), where c = LambertW(1/2) = 0.351733711249195826... (see A202356). - Vaclav Kotesovec, Jan 10 2014

Example

E.g.f: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...
log(A(x))/x = 1 + x*A(x) + x^2*A(x)^2 + x^3*A(x)^3 + x^4*A(x)^4 +...

Mathematica

Table[Sum[n! * (n-k+1)^(k-1)/k! * Binomial[n-1, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 10 2014 *)

Prog

(PARI) {a(n, m=1)=if(n==0, 1, sum(k=0, n, n!/k!*m*(n-k+m)^(k-1)*binomial(n-1, n-k)))}
(PARI) {a(n, m=1)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(x/(1-x*A))); n!*polcoeff(A^m, n)}

Crossrefs

Cf. A161633 (e.g.f. = log(A(x))/x).

Cf. A212722, A212917, A245265, A125500.

Sequence in context: A306576 A303064 A343672 * A371320 A121083 A343685

Adjacent sequences: A161627 A161628 A161629 * A161631 A161632 A161633

Keyword

nonn

Author

Paul D. Hanna, Jun 17 2009

Status

approved