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 A052873 #48 Mar 15 2025 11:29:19
%S A052873 1,1,5,46,629,11496,263857,7301680,236748969,8806142080,369714769181,
%T A052873 17296339048704,892335712777885,50333180563864576,3081739132775658825,
%U A052873 203555129140352505856,14428195498061848405073,1092403962489972428144640,87990832863810814525250869
%N A052873 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 - x*A(x))).
%C A052873 Previous name was: A simple grammar.
%H A052873 Seiichi Manyama, <a href="/A052873/b052873.txt">Table of n, a(n) for n = 0..357</a>
%H A052873 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=844">Encyclopedia of Combinatorial Structures 844</a>.
%H A052873 R. Lorentz, S. Tringali, C. H. Yan, <a href="http://arxiv.org/abs/1511.04039">Generalized Goncarov polynomials</a>, arXiv preprint arXiv:1511.04039 [math.CO], 2015.
%F A052873 E.g.f.: exp(RootOf(exp(_Z)*x*_Z+exp(_Z)*x-_Z)).
%F A052873 1 = Sum_{n>=0} a(n)*exp((n+1)*x/(x-1))*x^n/n!. - _Vladeta Jovovic_, Jul 20 2005
%F A052873 a(n) = Sum_{k=0..n} (n+1)^(k-1)*n!/k!*binomial(n-1,k-1). - _Vladeta Jovovic_, Jul 02 2006
%F A052873 E.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n! / (1-x*A(x))^n. - _Paul D. Hanna_, Sep 08 2012
%F A052873 Equivalently:
%F A052873 E.g.f. satisfies: A(x) = exp( x*A(x)/(1 - x*A(x)) ). - _Olivier Gérard_, Dec 29 2013
%F A052873 a(n) ~ (sqrt(5)-1) * 2^(n-1/2) * n^(n-1) * exp((sqrt(5)-1 + (sqrt(5)-3)*n)/2) / (5^(1/4) * (3-sqrt(5))^(n+1/2)). - _Vaclav Kotesovec_, Jan 08 2014
%F A052873 a(n) = n!*hypergeom([1-n],[2],-n-1) for n >= 1. - _Peter Luschny_, Apr 20 2016
%F A052873 E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x) ) ). - _Seiichi Manyama_, Mar 15 2025
%p A052873 spec := [S,{C=Sequence(B,1 <= card),S=Set(C),B=Prod(Z,S)},labeled]:
%p A052873 seq(combstruct[count](spec,size=n), n=0..20);
%p A052873 # Alternatively:
%p A052873 a := n -> `if`(n=0,1, n!*hypergeom([1-n],[2],-n-1)):
%p A052873 seq(simplify(a(n)), n=0..16); # _Peter Luschny_, Apr 20 2016
%t A052873 Table[Sum[(n+1)^(k-1)*n!/k!*Binomial[n-1,k-1],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jan 08 2014 *)
%o A052873 (PARI) {a(n)=if(n==0,1,sum(k=0,n,(n+1)^(k-1)*n!/k!*binomial(n-1,k-1)))} \\ _Paul D. Hanna_, Sep 08 2012
%o A052873 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(m+1)^(m-1)*x^m/m!/(1-x*A+x*O(x^n))^m));n!*polcoeff(A,n)} \\ _Paul D. Hanna_, Sep 08 2012
%Y A052873 Cf. A052868, A088690, A161630.
%Y A052873 Cf. A382036, A382037, A382038.
%K A052873 easy,nonn
%O A052873 0,3
%A A052873 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052873 New name using e.g.f., _Vaclav Kotesovec_, Jan 08 2014