A052895 E.g.f.: (1/2)/(exp(x) - 1) * (1 - (5 - 4*exp(x))^(1/2)).

1, 1, 5, 43, 545, 9211, 195305, 4990483, 149371745, 5128125451, 198696086105, 8578228640323, 408387804764945, 21256203702751291, 1200890923560864905, 73191086773679576563, 4786857909878612350145, 334410103752029126714731

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

Previous name was: A simple grammar.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 871

Crossrefs

Cf. A000108, A006531, A251568.

Programs

Maple

spec := [S,{C=Set(Z,1 <= card),S=Sequence(B),B=Prod(C,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

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 *)
a[n_] = Sum[k! StirlingS2[n, k] CatalanNumber[k], {k, 0, n}];
Table[a[n], {n, 0, 17}] (* Peter Luschny, Jan 15 2018 *)

Formula

E.g.f.: (1/2)/(exp(x) - 1)*(1 - (5 - 4*exp(x))^(1/2)).

a(n) = Sum_{k=0..n} k!*Stirling2(n,k)*Catalan(k). - Vladimir Kruchinin, Sep 15 2010

a(n) ~ sqrt(10)*n^(n-1) / (exp(n)*(log(5/4))^(n-1/2)). - Vaclav Kotesovec, Sep 30 2013

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

From Peter Bala, Jan 15 2018: (Start)

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.

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)

O.g.f.: 1 + Sum_{k>=1} A000108(k)*Product_{r=1..k} r*x/(1 - r*x). - Petros Hadjicostas, Jun 12 2020

Extensions

New name using e.g.f. from Vaclav Kotesovec, Sep 30 2013