A245265 E.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^4)).

1, 1, 3, 37, 649, 15461, 471571, 17456041, 760880625, 38178439849, 2167446089251, 137359883836781, 9612722107574521, 736277501363180557, 61265207586681046131, 5503291392884323494961, 530778414439201798454881, 54706967800114521799571921, 6000952913613549583603208515

Offset: 0

Author

Vaclav Kotesovec, Jul 15 2014

Keywords

• nonn
• easy

Comments

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)).

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 37*x^3/3! + 649*x^4/4! + 15461*x^5/5! + 471571*x^6/6! + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Vaclav Kotesovec, Asymptotic of sequences A161630, A212722, A212917 and A245265

Crossrefs

Cf. A161630 (p=1), A212722 (p=2), A212917 (p=3).

Cf. A030178.

Cf. A366234 (log).

Programs

Mathematica

Table[Sum[n! * (1 + 4*(n-k))^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}]

PARI

for(n=0,30, print1(sum(k=0,n, n!*(1 + 4*(n-k))^(k-1)/k!*binomial(n-1,n-k)), ", ")) \\ G. C. Greubel, Nov 17 2017

Formula

a(n) = Sum_{k=0..n} n! * (1 + 4*(n-k))^(k-1)/k! * C(n-1,n-k).

a(n) ~ n^(n-1) * (1+2*LambertW(1))^(n+1/2) / (exp(n) * (LambertW(1))^(2*n) * (4*sqrt(1+LambertW(1)))). - Vaclav Kotesovec, Jul 15 2014