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 A052886 Expansion of e.g.f.: (1/2) - (1/2)*(5 - 4*exp(x))^(1/2). 13
 0, 1, 3, 19, 207, 3211, 64383, 1581259, 45948927, 1541641771, 58645296063, 2494091717899, 117258952478847, 6038838138717931, 338082244882740543, 20443414320405268939, 1327850160592214291967, 92200405122521276427691, 6815359767190023358085823, 534337135055010788022858379 (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 859. FORMULA E.g.f.: (1/2) - (1/2)*(5 - 4*exp(x))^(1/2). a(n) = 1 + Sum_{k=1..n-1} binomial(n,k)*a(k)*a(n-k). - Vladeta Jovovic, Feb 02 2005 a(n) = Sum_{k=1..n} k!*Stirling2(n,k)*C(k-1), where C(k) = Catalan numbers (A000108). - Vladimir Kruchinin, Sep 15 2010 a(n) ~ sqrt(5/2)/2 * n^(n-1) / (exp(n)*(log(5/4))^(n-1/2)). - Vaclav Kotesovec, Sep 30 2013 Equals the logarithmic derivative of A293379. - Paul D. Hanna, Oct 22 2017 O.g.f.: Sum_{k>=1} C(k-1)*Product_{r=1..k} r*x/(1-r*x), where C = A000108. - Petros Hadjicostas, Jun 12 2020 MAPLE spec := [S, {C=Set(Z, 1 <= card), S=Prod(B, C), B=Sequence(S)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20); MATHEMATICA CoefficientList[Series[1/2-1/2*(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 - 1], {k, 1, n}]; Array[a, 20, 0] (* Peter Luschny, Apr 30 2020 *) PROG (PARI) N=66; x='x+O('x^N); concat([0], Vec(serlaplace(serreverse(log(1+x-x^2))))) \\ Joerg Arndt, May 25 2011 (PARI) lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = 1+ sum(k=1, n-1, binomial(n, k)*va[k]*va[n-k]); ); concat(0, va); } \\ Michel Marcus, Apr 30 2020 CROSSREFS Cf. A000108, A054867, A293379. Sequence in context: A108993 A245308 A182956 * A180563 A294330 A079144 Adjacent sequences: A052883 A052884 A052885 * A052887 A052888 A052889 KEYWORD easy,nonn AUTHOR encyclopedia(AT)pommard.inria.fr, Jan 25 2000 EXTENSIONS New name using e.g.f. by Vaclav Kotesovec, Sep 30 2013 STATUS approved

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Last modified November 29 16:46 EST 2023. Contains 367445 sequences. (Running on oeis4.)