A371329 -id:A371329 - 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

A376036 E.g.f. satisfies A(x) = (exp(x / (1 - A(x))^3) - 1) / (1 - A(x)).

Original entry on oeis.org

0, 1, 9, 190, 6435, 301126, 18007161, 1311752590, 112703870439, 11158543451926, 1250964512674533, 156642117419304958, 21668625406445359227, 3281750147124057118966, 540094007004476783547825, 95975344500184607391266734, 18314947854834472094038237647

Offset: 0

Seiichi Manyama, Sep 07 2024

nonn

Index entries for reversions of series

Cf. A274265, A376034, A376035.

Cf. A371329, A376037.

a(n) = sum(k=1, n, (3*n+2*k-2)!/(3*n+k-1)!*stirling(n, k, 2));

a(n) = Sum_{k=1..n} (3*n+2*k-2)!/(3*n+k-1)! * Stirling2(n,k).

E.g.f.: Series_Reversion( (1 - x)^3 * log(1 + x * (1 - x)) ).

A376037 E.g.f. satisfies A(x) = (exp(x / (1 - A(x))^2) - 1) / (1 - A(x)).

Original entry on oeis.org

0, 1, 7, 115, 3047, 111771, 5244555, 299941195, 20239069807, 1574068019851, 138641219870243, 13640672949173403, 1482772864485867399, 176478769995088245595, 22825571074271407363771, 3187825736999237502879019, 478120273969744650293424095

Offset: 0

Seiichi Manyama, Sep 07 2024

nonn

Index entries for reversions of series

Cf. A371329, A376036.

Cf. A085527, A371371.

a(n) = sum(k=1, n, (2*n+2*k-2)!/(2*n+k-1)!*stirling(n, k, 2));

a(n) = Sum_{k=1..n} (2*n+2*k-2)!/(2*n+k-1)! * Stirling2(n,k).

E.g.f.: Series_Reversion( (1 - x)^2 * log(1 + x * (1 - x)) ).

A371330 E.g.f. satisfies A(x) = (exp(x/(1 - A(x))) - 1)/(1 - A(x))^2.

Original entry on oeis.org

0, 1, 7, 112, 2901, 104176, 4788191, 268323756, 17744075761, 1352623086136, 116780496526515, 11263219375425172, 1200239384528276285, 140044340185131990336, 17757626485468691645479, 2431398542489983741458940, 357522675169127219183137737

Offset: 0

Seiichi Manyama, Mar 19 2024

nonn

Index entries for reversions of series

Cf. A052892, A371329.

a(n) = sum(k=1, n, (n+3*k-2)!/(n+2*k-1)!*stirling(n, k, 2));

a(n) = Sum_{k=1..n} (n+3*k-2)!/(n+2*k-1)! * Stirling2(n,k).

E.g.f.: Series_Reversion( (1 - x) * log(1 + x * (1 - x)^2) ). - Seiichi Manyama, Sep 08 2024

cp's OEIS Frontend

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

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A376036 E.g.f. satisfies A(x) = (exp(x / (1 - A(x))^3) - 1) / (1 - A(x)).

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A376037 E.g.f. satisfies A(x) = (exp(x / (1 - A(x))^2) - 1) / (1 - A(x)).

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A371330 E.g.f. satisfies A(x) = (exp(x/(1 - A(x))) - 1)/(1 - A(x))^2.

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