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A052895 E.g.f.: (1/2)/(exp(x)-1)*(1-(5-4*exp(x))^(1/2)). 6
1, 1, 5, 43, 545, 9211, 195305, 4990483, 149371745, 5128125451, 198696086105, 8578228640323, 408387804764945, 21256203702751291, 1200890923560864905, 73191086773679576563, 4786857909878612350145, 334410103752029126714731 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

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)

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 *)

CROSSREFS

Cf. A000108, A006531, A251568.

Sequence in context: A256033 A251568 A090470 * A162695 A161635 A005989

Adjacent sequences:  A052892 A052893 A052894 * A052896 A052897 A052898

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified October 19 00:36 EDT 2018. Contains 316327 sequences. (Running on oeis4.)