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Expansion of e.g.f.: (1/2) - (1/2)*(5 - 4*exp(x))^(1/2).
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%I #68 Sep 09 2024 09:33:25

%S 0,1,3,19,207,3211,64383,1581259,45948927,1541641771,58645296063,

%T 2494091717899,117258952478847,6038838138717931,338082244882740543,

%U 20443414320405268939,1327850160592214291967,92200405122521276427691,6815359767190023358085823,534337135055010788022858379

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

%C Previous name was: A simple grammar.

%C 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 Vincenzo Librandi, <a href="/A052886/b052886.txt">Table of n, a(n) for n = 0..200</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=859">Encyclopedia of Combinatorial Structures 859</a>.

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

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

%F a(n) = 1 + Sum_{k=1..n-1} binomial(n,k)*a(k)*a(n-k). - _Vladeta Jovovic_, Feb 02 2005

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

%F Equals the logarithmic derivative of A293379. - _Paul D. Hanna_, Oct 22 2017

%F 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 a(n) = A366377(A000040(n)) = A366884(A098719(n)). - _Antti Karttunen_, Jan 02 2024

%F From _Seiichi Manyama_, Sep 09 2024: (Start)

%F E.g.f. satisfies A(x) = (exp(x) - 1) / (1 - A(x)).

%F E.g.f.: Series_Reversion( log(1 + x * (1 - x)) ). (End)

%p 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 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 a[n_] := Sum[k! StirlingS2[n, k] CatalanNumber[k - 1], {k, 1, n}];

%t Array[a, 20, 0] (* _Peter Luschny_, Apr 30 2020 *)

%o (PARI) N=66; x='x+O('x^N);

%o concat([0],Vec(serlaplace(serreverse(log(1+x-x^2)))))

%o \\ _Joerg Arndt_, May 25 2011

%o (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 (PARI)

%o A000108(n) = binomial(2*n, n)/(n+1);

%o A052886(n) = sum(k=1,n,k!*stirling(n,k,2)*A000108(k-1)); \\ _Antti Karttunen_, Jan 02 2024

%Y Cf. A000108, A054867, A277120, A293379, A366377, A366884.

%Y Cf. A371316, A371317.

%Y Cf. A371329, A376036, A376037.

%K easy,nonn

%O 0,3

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

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