A293379 -id:A293379 - cp's OEIS frontend

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

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

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

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

Cf. A000108, A054867, A277120, A293379, A366377, A366884.
Cf. A371316, A371317.
Cf. A371329, A376036, A376037.

spec := [S,{C=Set(Z,1 <= card),S=Prod(B,C),B=Sequence(S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

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 *)

N=66; x='x+O('x^N);
concat([0],Vec(serlaplace(serreverse(log(1+x-x^2)))))
\\ Joerg Arndt, May 25 2011

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

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

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)

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

A294115 G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where Sum_{n>=1} L(n) * x^n/n! = Series_Reversion( log(1 + x/A(x)) ).

Original entry on oeis.org

1, 1, 2, 10, 143, 5959, 904224, 696895088, 3563009122225, 144004257475683137, 52273888783668336094726, 189699379891906830471022186526, 7572226826806850232281722700245568807, 3627110408773444347271222282038547230122245455, 22586092882428159778440302586299616247303225297287979548, 1969016989037466758104728399066094312610056241493227691736998060636, 2574833047387344521023398134994106823445574761658761070132072595536874966252691

Offset: 0

Paul D. Hanna, Oct 22 2017

nonn

This sequence is motivated by the following conjectures:
(C1) Given integer series G(x) such that G(0) = G'(0) = 1, define L(n) by
Sum_{n>=1} L(n) * x^n/n! = Series_Reversion( log(G(x)) )
then exp( Sum_{n>=1} L(n) * x^n/n ) is also an integer series;
(C2) Given G(x) = 1 + x*G(x)^m, define L(n) by
Sum_{n>=1} L(n) * x^n/n! = Series_Reversion( log(G(x)) )
then exp( Sum_{n>=1} L(n) * x^n/n ) = (1 + m*x)/(1 + (m-1)*x).

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 143*x^4 + 5959*x^5 + 904224*x^6 + 696895088*x^7 + 3563009122225*x^8 + 144004257475683137*x^9 +...
The logarithm of the g.f. begins
log(A(x)) = x + 3*x^2/2 + 25*x^3/3 + 531*x^4/4 + 29041*x^5/5 + 5388603*x^6/6 + 4871887945*x^7/7 + 28498490189571*x^8/8 + 1296006243863566561*x^9/9 +...+ L(n)*x^n/n +...
such that the same coefficients L(n) are also found in
Series_Reversion( log(1 + x/A(x)) ) = x + 3*x^2/2! + 25*x^3/3! + 531*x^4/4! + 29041*x^5/5! + 5388603*x^6/6! + 4871887945*x^7/7! + 28498490189571*x^8/8! +...+ L(n)*x^n/n! +...

Paul D. Hanna, Table of n, a(n) for n = 0..50

Cf. A293379.

{a(n) = my(A=1+x, L); for(i=0,n, L = x*serlaplace( 1/x*serreverse( log(1 + x/A +O(x^(n+2))) ) ); A = exp(L);); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

cp's OEIS Frontend

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

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A294115 G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where Sum_{n>=1} L(n) * x^n/n! = Series_Reversion( log(1 + x/A(x)) ).

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cp's OEIS Frontend

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

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Author

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Programs

Maple

Mathematica

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PARI

PARI

Formula

Extensions

A294115 G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where Sum_{n>=1} L(n) * x^n/n! = Series_Reversion( log(1 + x/A(x)) ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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