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A052895 E.g.f.: (1/2)/(exp(x)-1)*(1-(5-4*exp(x))^(1/2)). 6

%I

%S 1,1,5,43,545,9211,195305,4990483,149371745,5128125451,198696086105,

%T 8578228640323,408387804764945,21256203702751291,1200890923560864905,

%U 73191086773679576563,4786857909878612350145,334410103752029126714731

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

%C Previous name was: A simple grammar.

%H Vincenzo Librandi, <a href="/A052895/b052895.txt">Table of n, a(n) for n = 0..200</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=871">Encyclopedia of Combinatorial Structures 871</a>

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

%F a(n) = Sum_{k=0..n} k!*Stirling2(n,k)*Catalan(k). - _Vladimir Kruchinin_, Sep 15 2010

%F a(n) ~ sqrt(10)*n^(n-1) / (exp(n)*(log(5/4))^(n-1/2)). - _Vaclav Kotesovec_, Sep 30 2013

%F 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 From _Peter Bala_, Jan 15 2018: (Start)

%F 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 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)

%p 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 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 a[n_] = Sum[k! StirlingS2[n, k] CatalanNumber[k], {k, 0, n}];

%t Table[a[n], {n, 0, 17}] (* _Peter Luschny_, Jan 15 2018 *)

%Y Cf. A000108, A006531, A251568.

%K easy,nonn

%O 0,3

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

%E New name using e.g.f. from _Vaclav Kotesovec_, Sep 30 2013

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Last modified November 21 06:00 EST 2019. Contains 329350 sequences. (Running on oeis4.)