This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A161633 #19 Oct 07 2024 01:06:29
%S A161633 1,1,4,27,268,3525,57966,1146061,26500552,702069129,20974309210,
%T A161633 697754762001,25584428686620,1025230366195789,44579963354153878,
%U A161633 2090676600895922565,105191995364927688976,5652501986238910061073,323083811850594613809714,19573120681427758058921881
%N A161633 E.g.f. satisfies A(x) = 1/(1 - x*exp(x*A(x))).
%H A161633 Seiichi Manyama, <a href="/A161633/b161633.txt">Table of n, a(n) for n = 0..374</a>
%F A161633 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A161633 (1) A(x) = 1 + x*A(x)*exp(x*A(x)).
%F A161633 (2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(x)) ).
%F A161633 (3) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-1)*x*A(x)) for all fixed nonnegative m.
%F A161633 a(n) = n! * Sum_{k=0..n} binomial(n+1,k)/(n+1) * k^(n-k)/(n-k)!.
%F A161633 Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then a(n,m) = n! * Sum_{k=0..n} binomial(n+m,k)*m/(n+m) * k^(n-k)/(n-k)!.
%F A161633 a(n) ~ n^(n-1) * c * ((c-1)*c)^(n+1/2) / (sqrt(2*c-1) * exp(n)), where c = 1 + 1/(2*LambertW(1/2)) = 2.4215299358831166... - _Vaclav Kotesovec_, Jan 10 2014
%e A161633 E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 268*x^4/4! + 3525*x^5/5! +...
%e A161633 exp(x*A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...
%t A161633 Flatten[{1,Table[n!*Sum[Binomial[n+1,k]/(n+1) * k^(n-k)/(n-k)!,{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Jan 10 2014 *)
%o A161633 (PARI) a(n,m=1)=n!*sum(k=0,n,binomial(n+m,k)*m/(n+m)*k^(n-k)/(n-k)!)
%Y A161633 Cf. A006153, A161630 (e.g.f. = exp(x*A(x))), A213644, A364980, A364981.
%Y A161633 Cf. A366232, A366233, A366234, A366235.
%K A161633 nonn
%O A161633 0,3
%A A161633 _Paul D. Hanna_, Jun 18 2009