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 A052895 E.g.f.: (1/2)/(exp(x) - 1) * (1 - (5 - 4*exp(x))^(1/2)). 7
 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) O.g.f.: 1 + Sum_{k>=1} A000108(k)*Product_{r=1..k} r*x/(1 - r*x). - Petros Hadjicostas, Jun 12 2020 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 December 5 00:42 EST 2020. Contains 338943 sequences. (Running on oeis4.)