A212917 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^3) ).

1, 1, 3, 31, 469, 9681, 254701, 8131999, 305626329, 13218345793, 646712664121, 35315446759671, 2129341219106773, 140506900034640049, 10071589943109973461, 779311468200041101711, 64742128053980794659121, 5747587082198264156035329, 543023929087191507383612785

Offset

0,3

Comments

From Vaclav Kotesovec, Jul 15 2014: (Start)

Generally, if e.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^p)), p>=1, then

r = 4*LambertW(sqrt(p)/2)^2 / (p*(1+2*LambertW(sqrt(p)/2))),

A(r) = (sqrt(p)/(2*LambertW(sqrt(p)/2)))^(2/p),

a(n) ~ p^(n-1+1/p) * (1+2*LambertW(sqrt(p)/2))^(n+1/2) * n^(n-1) / (sqrt(1+LambertW(sqrt(p)/2)) * exp(n) * 2^(2*n+2/p) * LambertW(sqrt(p)/2)^(2*n+2/p-1/2)).

(End)

Links

G. C. Greubel, Table of n, a(n) for n = 0..349

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

Formula

a(n) = Sum_{k=0..n} n! * (1 + 3*(n-k))^(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*(m + 3*(n-k))^(k-1)/k! * C(n-1,n-k).

a(n) ~ 3^(n-2/3) * n^(n-1) * (1+2*c)^(n+1/2) / (sqrt(1+c) * 2^(2*n+2/3) * exp(n) * c^(2*n+1/6)), where c = LambertW(sqrt(3)/2) = 0.5166154518588324282494... . - Vaclav Kotesovec, Jul 15 2014

Example

E.g.f: A(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 469*x^4/4! + 9681*x^5/5! + ...
such that, by definition:
log(A(x))/x = 1 + x*A(x)^3 + x^2*A(x)^6 + x^3*A(x)^9 + x^4*A(x)^12 + ...
Related expansions:
log(A(x)) = x/(1-x*A(x)^3) = x + 2*x^2/2! + 24*x^3/3! + 348*x^4/4! + 7140*x^5/5! + 186750*x^6/6! + ... + n*A366233(n-1)*x^n/n! + ...
A(x)^3 = 1 + 3*x + 15*x^2/2! + 153*x^3/3! + 2421*x^4/4! + 51363*x^5/5! + 1375029*x^6/6! + ...
A(x)^6 = 1 + 6*x + 48*x^2/2! + 576*x^3/3! + 9864*x^4/4! + 221256*x^5/5! + 6156756*x^6/6! + ...

Mathematica

Table[Sum[n! * (1 + 3*(n-k))^(k-1)/k! * Binomial[n-1, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 15 2014 *)

Prog

(PARI) {a(n, m=1)=if(n==0, 1, sum(k=0, n, n!/k!*m*(m+3*(n-k))^(k-1)*binomial(n-1, n-k)))}
(PARI) {a(n, m=1)=local(A=1+x); for(i=1, n, A=exp(x/(1-x*A^3+x*O(x^n)))); n!*polcoeff(A^m, n)}
for(n=0, 21, print1(a(n), ", "))

Crossrefs

Cf. A161630, A212722, A245265.

Cf. A366233 (log).

Sequence in context: A349038 A126346 A142999 * A223993 A342206 A346313

Adjacent sequences: A212914 A212915 A212916 * A212918 A212919 A212920

Keyword

nonn

Author

Paul D. Hanna, May 30 2012

Status

approved