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 A162655 #10 Nov 22 2021 02:27:43
%S A162655 1,1,4,33,416,7100,153234,4004000,122919208,4336955424,172946624880,
%T A162655 7692618593352,377615317473624,20278301717340888,1182581903027279832,
%U A162655 74428445506232769240,5028336618916834615104,362962785521720282899200
%N A162655 E.g.f. satisfies: A(x) = (1 + x*A(x))^A(x).
%C A162655 Contribution from _Paul D. Hanna_, Jul 19 2009: (Start)
%C A162655 More generally, if G(x) = (1 + x*G(x)^p)^(G(x)^q), then
%C A162655 [x^n/n! ] G(x)^m = Sum_{k=0..n} m*(pn+qk+m)^(k-1) * Stirling1(n,k), and
%C A162655 [x^n/n! ] log(G(x)) = Sum_{k=1..n} (pn+qk)^(k-1) * Stirling1(n,k). (End)
%H A162655 Seiichi Manyama, <a href="/A162655/b162655.txt">Table of n, a(n) for n = 0..359</a>
%F A162655 (1) a(n) = Sum_{k=0..n} (n+k+1)^(k-1) * Stirling1(n,k).
%F A162655 Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
%F A162655 (2) a(n,m) = Sum_{k=0..n} m*(n+k+m)^(k-1) * Stirling1(n,k) ;
%F A162655 which is equivalent to the following:
%F A162655 (3) a(n,m) = Sum_{k=0..n} m*(n+k+m)^(k-1) * {[x^(n-k)] Product_{j=1..n-1} (1-j*x) };
%F A162655 (4) a(n,m) = n!*Sum_{k=0..n} m*(n+k+m)^(k-1) * {[x^(n-k)] (log(1+x)/x)^k/k!}.
%F A162655 a(n) ~ s^2*sqrt(r*(1+r*s)/(1+r*s*(1+s)*(2+r*s))) * n^(n-1) / (exp(n)*r^n), where r = 0.21551711955114319212... and s = 1.7128732151580576508... are roots of the system of equations s*(r*s/(1+r*s) + log(1+r*s)) = 1, (1+r*s)^s = s. - _Vaclav Kotesovec_, Jul 15 2014
%e A162655 E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 416*x^4/4! + 7100*x^5/5! +...
%e A162655 log(A(x)) = A(x)*log(1 + x*A(x)) where
%e A162655 log(A(x)) = x + 3*x^2/2! + 23*x^3/3! + 278*x^4/4! + 4624*x^5/5! + 98064*x^6/6! +...
%e A162655 log(1 + x*A(x)) = x + x^2/2! + 8*x^3/3! + 90*x^4/4! + 1444*x^5/5! + 29880*x^6/6! +...
%t A162655 Table[Sum[(n+k+1)^(k-1) * StirlingS1[n,k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jul 15 2014 *)
%o A162655 (PARI) {a(n,m=1)=sum(k=0,n,m*(n+k+m)^(k-1)*polcoeff(prod(j=1,n-1,1-j*x),n-k))}
%o A162655 (PARI) {a(n,m=1)=sum(k=0,n,m*(n+k+m)^(k-1)*n!/k!*polcoeff((log(1+x+x*O(x^n))/x)^k,n-k))}
%o A162655 (PARI) a(n,m=1)=sum(k=0,n,m*(n+k+m)^(k-1)*stirling(n,k,1));
%Y A162655 Cf. A008275 (Stirling1), A141209 (variant).
%Y A162655 Cf. A162863. [From _Paul D. Hanna_, Jul 19 2009]
%K A162655 nonn
%O A162655 0,3
%A A162655 _Paul D. Hanna_, Jul 08 2009