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A052886 Expansion of e.g.f.: (1/2) - (1/2)*(5 - 4*exp(x))^(1/2).

%I A052886 #68 Sep 09 2024 09:33:25
%S A052886 0,1,3,19,207,3211,64383,1581259,45948927,1541641771,58645296063,
%T A052886 2494091717899,117258952478847,6038838138717931,338082244882740543,
%U A052886 20443414320405268939,1327850160592214291967,92200405122521276427691,6815359767190023358085823,534337135055010788022858379
%N A052886 Expansion of e.g.f.: (1/2) - (1/2)*(5 - 4*exp(x))^(1/2).
%C A052886 Previous name was: A simple grammar.
%C A052886 From the symmetry present in _Vladeta Jovovic_'s Feb 02 2005 formula, it is easy to see that there are no positive even numbers in this sequence. Question: are there any multiples of 5 after the initial zero? Compare also to the comments in A366884. - _Antti Karttunen_, Jan 02 2024
%H A052886 Vincenzo Librandi, <a href="/A052886/b052886.txt">Table of n, a(n) for n = 0..200</a>
%H A052886 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=859">Encyclopedia of Combinatorial Structures 859</a>.
%H A052886 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A052886 E.g.f.: (1/2) - (1/2)*(5 - 4*exp(x))^(1/2).
%F A052886 a(n) = 1 + Sum_{k=1..n-1} binomial(n,k)*a(k)*a(n-k). - _Vladeta Jovovic_, Feb 02 2005
%F A052886 a(n) = Sum_{k=1..n} k!*Stirling2(n,k)*C(k-1), where C(k) = Catalan numbers (A000108). - _Vladimir Kruchinin_, Sep 15 2010
%F A052886 a(n) ~ sqrt(5/2)/2 * n^(n-1) / (exp(n)*(log(5/4))^(n-1/2)). - _Vaclav Kotesovec_, Sep 30 2013
%F A052886 Equals the logarithmic derivative of A293379. - _Paul D. Hanna_, Oct 22 2017
%F A052886 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
%F A052886 a(n) = A366377(A000040(n)) = A366884(A098719(n)). - _Antti Karttunen_, Jan 02 2024
%F A052886 From _Seiichi Manyama_, Sep 09 2024: (Start)
%F A052886 E.g.f. satisfies A(x) = (exp(x) - 1) / (1 - A(x)).
%F A052886 E.g.f.: Series_Reversion( log(1 + x * (1 - x)) ). (End)
%p A052886 spec := [S,{C=Set(Z,1 <= card),S=Prod(B,C),B=Sequence(S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052886 CoefficientList[Series[1/2-1/2*(5-4*E^x)^(1/2), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2013 *)
%t A052886 a[n_] := Sum[k! StirlingS2[n, k] CatalanNumber[k - 1], {k, 1, n}];
%t A052886 Array[a, 20, 0] (* _Peter Luschny_, Apr 30 2020 *)
%o A052886 (PARI) N=66; x='x+O('x^N);
%o A052886 concat([0],Vec(serlaplace(serreverse(log(1+x-x^2)))))
%o A052886 \\ _Joerg Arndt_, May 25 2011
%o A052886 (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
%o A052886 (PARI)
%o A052886 A000108(n) = binomial(2*n, n)/(n+1);
%o A052886 A052886(n) = sum(k=1,n,k!*stirling(n,k,2)*A000108(k-1)); \\ _Antti Karttunen_, Jan 02 2024
%Y A052886 Cf. A000108, A054867, A277120, A293379, A366377, A366884.
%Y A052886 Cf. A371316, A371317.
%Y A052886 Cf. A371329, A376036, A376037.
%K A052886 easy,nonn
%O A052886 0,3
%A A052886 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052886 New name using e.g.f. by _Vaclav Kotesovec_, Sep 30 2013