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

0, 1, 3, 19, 207, 3211, 64383, 1581259, 45948927, 1541641771, 58645296063, 2494091717899, 117258952478847, 6038838138717931, 338082244882740543, 20443414320405268939, 1327850160592214291967, 92200405122521276427691, 6815359767190023358085823, 534337135055010788022858379

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Previous name was: A simple grammar.

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 859.
• Index entries for reversions of series

Crossrefs

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

Cf. A371316, A371317.

Cf. A371329, A376036, A376037.

Programs

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 *)

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

PARI

A000108(n) = binomial(2*n, n)/(n+1);
A052886(n) = sum(k=1,n,k!*stirling(n,k,2)*A000108(k-1)); \\ Antti Karttunen, Jan 02 2024

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

a(n) = A366377(A000040(n)) = A366884(A098719(n)). - Antti Karttunen, Jan 02 2024

From Seiichi Manyama, Sep 09 2024: (Start)

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

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

Extensions

New name using e.g.f. by Vaclav Kotesovec, Sep 30 2013