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 A143740 #31 Feb 16 2025 08:33:08
%S A143740 1,1,2,7,34,216,1696,15898,173468,2161036,30282076,471599316,
%T A143740 8082816160,151218316120,3066890630168,67031194526416,
%U A143740 1570793031033616,39290173530686544,1044871388684004304,29440090627527552976
%N A143740 E.g.f. satisfies A(x) = exp(x + x^2/2 * A(x)).
%H A143740 Seiichi Manyama, <a href="/A143740/b143740.txt">Table of n, a(n) for n = 0..419</a>
%H A143740 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A143740 E.g.f.: A(x) = -2*LambertW( -x^2*exp(x)/2 )/x^2.
%F A143740 E.g.f.: A(x) = Sum_{n>=0} (n+1)^(n-1)*(x^2/2)^n*exp((n+1)*x)/n!.
%F A143740 a(n) ~ sqrt(1+LambertW(1/sqrt(2*exp(1)))) * n^(n-1) /(2^(n+1/2) * exp(n) * (LambertW(1/sqrt(2*exp(1))))^(n+2)). - _Vaclav Kotesovec_, Jul 09 2013
%F A143740 Recurrence: a(0)=1, a(1)=1, for n > 1, a(n) = a(n-1) + Sum_{k=0..n-2} (n-1)*(n-k)*binomial(n-2,k)*a(k)*a(n-2-k)/2. - _Tani Akinari_, Nov 01 2017
%F A143740 From _Seiichi Manyama_, Apr 20 2023: (Start)
%F A143740 E.g.f.: exp(x - LambertW(-x^2/2 * exp(x))).
%F A143740 a(n) = n! * Sum_{k=0..floor(n/2)} (1/2)^k * (k+1)^(n-k-1) / (k! * (n-2*k)!). (End)
%e A143740 E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 34*x^4/4! + 216*x^5/5! + ...
%t A143740 CoefficientList[Series[-2*LambertW[-x^2*E^x/2]/x^2, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jul 09 2013 *)
%o A143740 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x+x^2*A/2));(n+0)!*polcoeff(A,n)}
%o A143740 (PARI) {a(n)=local(A=sum(m=0,n,(m+1)^(m-1)*(x^2/2)^m*exp((m+1)*x+x*O(x^n))/m!)); n!*polcoeff(A,n)}
%o A143740 (PARI) {a(n)=local(Ex=exp(x+x*O(x^n)),W=Ex);for(k=0,n,W=exp(x*W)); n!*polcoeff(subst(W,x,x^2*Ex/2)*Ex,n)}
%o A143740 (Maxima) a[n]:=(if n<2 then 1 else a[n-1]+sum((n-1)*(n-k)*binomial(n-2,k)*a[k]*a[n-2-k],k,0,n-2)/2);
%o A143740 makelist(a[n],n,0,100); /* _Tani Akinari_, Nov 01 2017 */
%Y A143740 Column k=1 of A362377.
%Y A143740 Cf. A125500.
%K A143740 nonn
%O A143740 0,3
%A A143740 _Paul D. Hanna_, Aug 30 2008