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 A052895 #41 Jun 13 2020 02:40:12
%S A052895 1,1,5,43,545,9211,195305,4990483,149371745,5128125451,198696086105,
%T A052895 8578228640323,408387804764945,21256203702751291,1200890923560864905,
%U A052895 73191086773679576563,4786857909878612350145,334410103752029126714731
%N A052895 E.g.f.: (1/2)/(exp(x) - 1) * (1 - (5 - 4*exp(x))^(1/2)).
%C A052895 Previous name was: A simple grammar.
%H A052895 Vincenzo Librandi, <a href="/A052895/b052895.txt">Table of n, a(n) for n = 0..200</a>
%H A052895 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=871">Encyclopedia of Combinatorial Structures 871</a>
%F A052895 E.g.f.: (1/2)/(exp(x) - 1)*(1 - (5 - 4*exp(x))^(1/2)).
%F A052895 a(n) = Sum_{k=0..n} k!*Stirling2(n,k)*Catalan(k). - _Vladimir Kruchinin_, Sep 15 2010
%F A052895 a(n) ~ sqrt(10)*n^(n-1) / (exp(n)*(log(5/4))^(n-1/2)). - _Vaclav Kotesovec_, Sep 30 2013
%F A052895 E.g.f.: 1/(1 + (1 - exp(x))/(1 + (1 - exp(x))/(1 + (1 - exp(x))/(1 + (1 - exp(x))/(1 + ...))))), a continued fraction. - _Ilya Gutkovskiy_, Nov 18 2017
%F A052895 From _Peter Bala_, Jan 15 2018: (Start)
%F A052895 E.g.f.: C(exp(x) - 1), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for A000108. Cf. A006531.
%F A052895 Conjecture: for fixed k = 1,2,..., the sequence a(n) (mod k) is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010. For example, modulo 10 the sequence becomes (1, 1, 5, 3, 5, 1, 5, 3, 5, ...), with an apparent period 1, 5, 3, 5 of length 4 = phi(10) beginning at a(1). (End)
%F A052895 O.g.f.: 1 + Sum_{k>=1} A000108(k)*Product_{r=1..k} r*x/(1 - r*x). - _Petros Hadjicostas_, Jun 12 2020
%p A052895 spec := [S,{C=Set(Z,1 <= card),S=Sequence(B),B=Prod(C,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052895 CoefficientList[Series[(1/2)/(E^x-1)*(1-(5-4*E^x)^(1/2)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2013 *)
%t A052895 a[n_] = Sum[k! StirlingS2[n, k] CatalanNumber[k], {k, 0, n}];
%t A052895 Table[a[n], {n, 0, 17}] (* _Peter Luschny_, Jan 15 2018 *)
%Y A052895 Cf. A000108, A006531, A251568.
%K A052895 easy,nonn
%O A052895 0,3
%A A052895 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052895 New name using e.g.f. from _Vaclav Kotesovec_, Sep 30 2013