A245265 - OEIS

%I #24 Oct 08 2023 07:03:12

%S 1,1,3,37,649,15461,471571,17456041,760880625,38178439849,

%T 2167446089251,137359883836781,9612722107574521,736277501363180557,

%U 61265207586681046131,5503291392884323494961,530778414439201798454881,54706967800114521799571921,6000952913613549583603208515

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

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

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

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

%C 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)).

%H Vincenzo Librandi, <a href="/A245265/b245265.txt">Table of n, a(n) for n = 0..200</a>

%H Vaclav Kotesovec, <a href="/A245265/a245265.pdf">Asymptotic of sequences A161630, A212722, A212917 and A245265</a>

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

%F 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

%e 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! + ...

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

%o (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

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

%Y Cf. A030178.

%Y Cf. A366234 (log).

%K nonn,easy

%O 0,3

%A _Vaclav Kotesovec_, Jul 15 2014