A366884 -id:A366884 - 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

A277120 Number of branching factorizations of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 11, 1, 3, 3, 15, 1, 11, 1, 11, 3, 3, 1, 45, 2, 3, 5, 11, 1, 19, 1, 51, 3, 3, 3, 62, 1, 3, 3, 45, 1, 19, 1, 11, 11, 3, 1, 195, 2, 11, 3, 11, 1, 45, 3, 45, 3, 3, 1, 113, 1, 3, 11, 188, 3, 19, 1, 11, 3, 19, 1, 345, 1, 3, 11, 11, 3

Offset: 1

Michel Marcus, Oct 01 2016

nonn

Per the formula, a(n) = 1 at prime n. As the sequence extends, additional values become more frequent than 1. These values can be characterized, for example, a(n) = 19 is seen at n corresponding to A007304, a(n) = 3 is seen at n corresponding to A006881, a(n) = 113 is seen at n corresponding to A085987. - Bill McEachen, Dec 28 2023
From Antti Karttunen, Jan 02 2024: (Start)
The value of a(n) depends only on the prime signature of n. In other words, for all i, j >= 1, it holds that A101296(i) = A101296(j) => a(i) = a(j). Moreover, it seems that the converse proposition also holds, that for all i, j >= 1, a(i) = a(j) => A101296(i) = A101296(j), i.e., for each distinct prime signature there exists a distinct value of a(n). This has been empirically checked up to the first 21001 prime signatures in A025487 (see A366884), and can be proved if one can show that the latter sequence (equally: A366377) is injective. If this conjecture holds, it would imply an unlimited number of statements like those given in the previous comment (see the formula section of A101296).
Questions: Are there any terms of the form 10k+4 or 10k+6? What is the asymptotic density of terms of the form 10k+5 (those ending with digit "5")? Compare to the data shown in A366884.
For squarefree n > 1, a(n) is never even, and apparently, never a multiple of five. See comments in A052886.
(End)

			In this scheme, the following factorizations of 12 are counted as distinct: 12, 2 x 6, 2 x (2 x 3), 2 x (3 x 2), 3 x 4, 3 x (2 x 2), 4 x 3, (2 x 2) x 3, 6 x 2, (2 x 3) x 2, (3 x 2) x 2, thus a(12) = 11. - _Antti Karttunen_, Nov 02 2016, based on the illustration given at page 14 of Knopfmacher & Mays paper.
The following factorizations of 30 are counted as distinct: 30, 2 x 15, 15 x 2, 3 x 10, 10 x 3, 5 x 6, 6 x 5, 2 x (3 x 5), 2 x (5 x 3), 3 x (2 x 5), 3 x (5 x 2), 5 x (2 x 3), 5 x (3 x 2), (2 x 3) x 5, (2 x 5) x 3, (3 x 2) x 5, (3 x 5) x 2, (5 x 2) x 3, (5 x 3) x 2, thus a(30) = 19. - _Antti Karttunen_, Jan 02 2024

Daniel Mondot, Table of n, a(n) for n = 1..9999
H.-K. Hwang, Théorèmes limites pour les structures combinatoires et les fonctions arithmétiques, PhD thesis, in Ecole Polytechnique, 1994. See page 198.
A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, 1(4):563-581,(2005). See page 14.
Index entries for sequences computed from exponents in factorization of n

Cf. A007317, A052886, A074206, A101296, A277130, A366377 [= a(A108951(n))], A366884 [= a(A025487(n))].

After n=1 differs from A104725 for the next time at n=32, where a(32) = 51, while A104725(32) = 52.

#include 
#define MAX 10000
/* Number of branching factorizations of n. */
unsigned long n, m, a, b, p, x, nbr[MAX];
int main(void)
{
  for (x=n=1; nDaniel Mondot, Oct 01 2016 */

v[n_] := v[n] = If[n == 1, 0, 1 + Sum[If[d == 1 || d^2 > n, 0, If[d^2 == n, 1, 2]*v[d]*v[n/d]], {d, Divisors[n]}]]; Table[v[n], {n, 1, 100}] (* Vaclav Kotesovec, Jan 13 2024, after Antti Karttunen *)

A277120(n) = if(1==n, 0, 1+sumdiv(n, d, if((1==d)||(d*d)>n,0,if((d*d)==n,1,2)*A277120(d)*A277120(n/d)))); \\ Antti Karttunen, Nov 02 2016, after Daniel Mondot's C-program above.

seq(n)={my(v=vector(n)); for(n=2, n, v[n] = 1 + sumdiv(n, d, v[d]*v[n/d])); v} \\ Andrew Howroyd, Nov 17 2018

a(1) = 0; for n > 1, a(n) = 1 + Sum_{d|n, 1 < d < n} a(d)*a(n/d). - Antti Karttunen, Nov 02 2016, after Daniel Mondot's C program, simplified Dec 30 2023.

For all n >= 1, a(prime^n) = A007317(n), and a(product of n distinct primes) = A052886(n). - Antti Karttunen, Dec 31 2023

More terms from Daniel Mondot, Oct 01 2016

A366377 Number of branching factorizations of the primorial inflation of n.

Original entry on oeis.org

0, 1, 3, 2, 19, 11, 207, 5, 62, 113, 3211, 45, 64383, 1709, 911, 15, 1581259, 345, 45948927, 645, 17753, 33797, 1541641771, 195, 9332, 822821, 2405, 12405, 58645296063, 6525, 2494091717899, 51, 428309, 23765093, 223031, 1890, 117258952478847, 793795349, 12293957, 3585, 6038838138717931, 154605, 338082244882740543, 296805

Offset: 1

Antti Karttunen, Dec 31 2023

nonn

Conjecture: Sequence is injective (no value occurs more than once). If true, then also the conjecture given in A277120 is correct. See also A366884.

Antti Karttunen, Table of n, a(n) for n = 1..121

Cf. A000040, A007317, A052886, A108951, A277120.

Permutation of A366884.

A002110(n) = prod(i=1,n,prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A002110(primepi(f[i, 1]))^f[i, 2]) }; \\ From A108951
memoA277120 = Map();
A277120(n) = if(1==n,0,my(v); if(mapisdefined(memoA277120,n,&v), v, v = 1+sumdiv(n,d,if((1==d)||(d*d)>n,0,if((d*d)==n,1,2)*A277120(d)*A277120(n/d))); mapput(memoA277120,n,v); (v)));
A366377(n) = A277120(A108951(n));

a(n) = A277120(A108951(n)).
a(n) = A366884(A329901(n)).
For n >= 1, a(2^n) = A007317(n), a(A000040(n)) = A052886(n).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions

A277120 Number of branching factorizations of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Mathematica

PARI

PARI

Formula

Extensions

A366377 Number of branching factorizations of the primorial inflation of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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