A277120 -id:A277120 - 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

A277130 Number of planar branching factorizations of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 6, 2, 3, 1, 14, 1, 3, 3, 24, 1, 14, 1, 14, 3, 3, 1, 78, 2, 3, 6, 14, 1, 25, 1, 112, 3, 3, 3, 110, 1, 3, 3, 78, 1, 25, 1, 14, 14, 3, 1, 464, 2, 14, 3, 14, 1, 78, 3, 78, 3, 3, 1, 206, 1, 3, 14, 568, 3, 25, 1, 14, 3, 25, 1, 850, 1, 3, 14, 14

Offset: 1

Michel Marcus, Oct 01 2016

nonn

A planar branching factorization of n is either the number n itself or a sequence of at least two planar branching factorizations, one of each factor in an ordered factorization of n. - Gus Wiseman, Sep 11 2018

			From _Gus Wiseman_, Sep 11 2018: (Start)
The a(12) = 14 planar branching factorizations:
  12,
  (2*6), (3*4), (4*3), (6*2), (2*2*3), (2*3*2), (3*2*2),
  (2*(2*3)), (2*(3*2)), (3*(2*2)), ((2*2)*3), ((2*3)*2), ((3*2)*2).
(End)

Daniel Mondot, Table of n, a(n) for n = 1..9999
A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, 1(4):563-581,(2005). See page 15.

Cf. A277120.
Cf. A000108, A001055, A074206, A118376, A281113, A319122, A319123.
Cf. A277130, A281118, A281119, A292504, A292505, A295279, A295281, A319136, A319137, A319138.

#include 
#include 
#include 
#define MAX 10000
/* Number of planar branching factorizations of n. */
#define lu unsigned long
lu nbr[MAX]; /* number of branching */
lu a, b, d, e; /* temporary variables */
lu n; lu m, p; // factors of n
lu x; // square root of n
void main(unsigned argc, char *argv[])
{
  memset(nbr, 0, MAX*sizeof(lu));
  for (b=0, n=1; nDaniel Mondot, May 19 2017 */

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
otfs[n_]:=Prepend[Join@@Table[Tuples[otfs/@f],{f,Select[ordfacs[n],Length[#]>1&]}],n];
Table[Length[otfs[n]],{n,20}] (* Gus Wiseman, Sep 11 2018 *)

a(prime^n) = A118376(n). a(product of n distinct primes) = A319122(n). - Gus Wiseman, Sep 11 2018

Terms a(65) and beyond from Daniel Mondot, May 19 2017

A339755 a(1) = 1; a(n+1) = 1 + Sum_{d|n} a(n/d) * a(d).

Original entry on oeis.org

1, 2, 5, 11, 27, 55, 131, 263, 571, 1168, 2445, 4891, 10113, 20227, 40979, 82229, 165632, 331265, 665365, 1330731, 2666729, 5334769, 10679319, 21358639, 42740683, 85482096, 171004645, 342015001, 684113793, 1368227587, 2736633741, 5473267483, 10946869669, 21893763789, 43788190107

Offset: 1

Ilya Gutkovskiy, Dec 15 2020

nonn

Cf. A038044, A068336, A097558, A122698, A277120, A325303.

a:= proc(n) option remember; uses numtheory;
      1+add(a(d)*a((n-1)/d), d=divisors(n-1))
    end:
seq(a(n), n=1..35);  # Alois P. Heinz, Dec 15 2020

a[1] = 1; a[n_] := a[n] = 1 + Sum[a[(n - 1)/d] a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 35}]

G.f.: x * (1/(1 - x) + Sum_{i>=1} Sum_{j>=1} a(i) * a(j) * x^(i*j)).

a(n) ~ c * 2^n, where c = 1.27442410710035207761153205319824525254716841098942446508584158048310907298... - Vaclav Kotesovec, Dec 16 2020

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).

A366884 Number of branching factorizations of the least integer of each prime signature (A025487).

Original entry on oeis.org

0, 1, 2, 3, 5, 11, 15, 45, 19, 51, 62, 195, 113, 188, 345, 873, 645, 731, 1890, 911, 3989, 207, 2405, 3585, 2950, 10221, 6525, 18483, 1709, 15775, 19569, 12235, 54718, 43545, 86515, 12405, 99215, 9332, 105447, 51822, 55885, 290611, 17753, 120075, 277203, 408105, 83505, 605135, 80565, 562739, 223191, 432975, 1533670

Offset: 1

Antti Karttunen, Jan 02 2024

nonn

Sequence appears to be injective, but can it be proved? This would prove also the conjectures given in A277120 and A366377.

Of the first 21001 terms, there are 701 terms ending with digit "0", 614 with "1", 68 with "2", 570 with "3", 0 with "4", 17795 with "5", 0 with "6", 550 with "7", 67 with "8", and 636 with "9". Why such an overrepresentation (~ 85% of the total) of the terms of form 10k+5? Do any terms of the form 10k+4 or 10k+6 exist? See also the comments in A052886.

Antti Karttunen, Table of n, a(n) for n = 1..21001
Michael De Vlieger, Plot k = a(n) mod 10 at (x,y) = (n mod 144, 1 + floor(n/144)), n = 1..20736, showing 0 in black, 1 in red, 2 in orange, 3 in yellow, 4 = dark green, 5 = bright green, 6 = cyan, 7 = light blue, 8 = dark blue, and 9 in purple.

Cf. A007317, A025487, A025488, A052886, A098719, A181815, A277120.

Permutation of A366377.

a(n) = A277120(A025487(n)).
a(n) = A366377(A181815(n)).
For all n >= 1, a(A025488(n)) = A007317(n), a(A098719(n)) = A052886(n).

A351787 a(1) = 1; a(n+1) = a(n) + Sum_{d|n} a(n/d) * a(d).

Original entry on oeis.org

1, 2, 6, 18, 58, 174, 546, 1638, 4986, 14994, 45214, 135642, 407838, 1223514, 3672726, 11018874, 33063498, 99190494, 297593514, 892780542, 2678403690, 8035217622, 24105833722, 72317501166, 216953071986, 650859219322, 1952579289318, 5857737927786, 17573218697070

Offset: 1

Ilya Gutkovskiy, Feb 19 2022

nonn

Cf. A038044, A084978, A122698, A277120, A339755, A341697, A345139, A351788.

a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[(n - 1)/d] a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 29}]

G.f.: x * ( 1 + Sum_{i>=1} Sum_{j>=1} a(i) * a(j) * x^(i*j) ) / (1 - x).

A351788 a(1) = 1; a(n) = a(n-1) + Sum_{d|n, 1 < d < n} a(n/d) * a(d).

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 8, 9, 13, 13, 25, 25, 33, 37, 57, 57, 83, 83, 117, 125, 151, 151, 233, 237, 287, 305, 387, 387, 503, 503, 649, 675, 789, 805, 1073, 1073, 1239, 1289, 1607, 1607, 1955, 1955, 2309, 2419, 2721, 2721, 3465, 3481, 4007, 4121, 4795, 4795, 5643, 5695

Offset: 1

Ilya Gutkovskiy, Feb 19 2022

nonn

Cf. A038044, A084978, A122698, A277120, A339755, A341697, A345139, A351787.

a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[If[1 < d < n, a[n/d] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]

G.f.: ( x + Sum_{i>=2} Sum_{j>=2} a(i) * a(j) * x^(i*j) ) / (1 - x).

A144757 Number of factor trees for n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 2, 5, 1, 6, 1, 6, 2, 2, 1, 20, 1, 2, 2, 6, 1, 12, 1, 14, 2, 2, 2, 30, 1, 2, 2, 20, 1, 12, 1, 6, 6, 2, 1, 70, 1, 6, 2, 6, 1, 20, 2, 20, 2, 2, 1, 60, 1, 2, 6, 42, 2, 12, 1, 6, 2, 12, 1, 140, 1, 2, 6, 6, 2, 12, 1, 70, 5, 2, 1, 60, 2, 2, 2, 20, 1, 60, 2, 6, 2, 2, 2, 252

Offset: 2

David Radcliffe, Sep 20 2008

A factor tree for n is a binary tree, with the root labeled with n and the terminal nodes labeled with primes, such that each non-terminal node is the product of its two child nodes. This is the number of prime factorizations of n, ignoring the commutativity and associativity of multiplication.

			a(12)=6 because 12 can be factored as (2*2)*3, (2*3)*2, (3*2)*2, 2*(2*3), 2*(3*2) and 3*(2*2).

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A000108, A001222, A008480, A277120.

a144757 n = a000108 (a001222 n - 1) * a008480 n
-- Reinhard Zumkeller, Nov 19 2015

a8480[n_] := With[{f = FactorInteger[n][[All, 2]]}, Total[f]!/Times @@ (f!)];
a[n_] := CatalanNumber[PrimeOmega[n] - 1] * a8480[n];
a /@ Range[2, 100] (* Jean-François Alcover, Sep 20 2019 *)

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

a(n)={if(n>=2, my(sig=factor(n)[,2]); my(t=vecsum(sig)-1); binomial(2*t, t)*t!/vecprod(apply(k->k!, sig)), 0)} \\ Andrew Howroyd, Nov 17 2018

a(n) = A000108(A001222(n)-1) * A008480(n).

A346116 a(1) = a(2) = 1; a(n+2) = 1 + Sum_{d|n} a(n/d) * a(d).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 11, 21, 23, 49, 51, 109, 103, 247, 207, 517, 435, 1086, 871, 2251, 1743, 4631, 3531, 9365, 7063, 19081, 14152, 38369, 28397, 77299, 56795, 155289, 113591, 311739, 227387, 624349, 454885, 1251509, 909771, 2504761, 1819955, 5014529, 3639911, 10033709, 7279823

Offset: 1

Ilya Gutkovskiy, Jul 05 2021

nonn

Cf. A144366, A277120, A339755.

a[1] = a[2] = 1; a[n_] := a[n] = 1 + Sum[a[(n - 2)/d] a[d], {d, Divisors[n - 2]}]; Table[a[n], {n, 1, 45}]

G.f.: x + x^2 * (1/(1 - x) + Sum_{i>=1} Sum_{j>=1} a(i) * a(j) * x^(i*j)).

A351797 a(1) = 1; a(n+1) = -a(n) + 2 * Sum_{d|n} a(n/d) * a(d).

Original entry on oeis.org

1, 1, 3, 9, 29, 87, 273, 819, 2493, 7497, 22607, 67821, 203919, 611757, 1836363, 5509437, 16531749, 49595247, 148796757, 446390271, 1339201845, 4017608811, 12052916861, 36158750583, 108476535993, 325429609661, 976289644659, 2928868963893, 8786609348535

Offset: 1

Ilya Gutkovskiy, Feb 19 2022

nonn

Cf. A038044, A084978, A122698, A277120, A339755, A341697, A345139, A351787, A351788.

a[1] = 1; a[n_] := a[n] = -a[n - 1] + 2 Sum[a[(n - 1)/d] a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 29}]

G.f.: x * ( 1 + 2 * Sum_{i>=1} Sum_{j>=2} a(i) * a(j) * x^(i*j) ) / (1 - x).

a(n) = A351787(n) / 2 for n > 1.

cp's OEIS Frontend

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

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A277130 Number of planar branching factorizations of n.

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A339755 a(1) = 1; a(n+1) = 1 + Sum_{d|n} a(n/d) * a(d).

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

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A366884 Number of branching factorizations of the least integer of each prime signature (A025487).

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A351787 a(1) = 1; a(n+1) = a(n) + Sum_{d|n} a(n/d) * a(d).

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A351788 a(1) = 1; a(n) = a(n-1) + Sum_{d|n, 1 < d < n} a(n/d) * a(d).

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A144757 Number of factor trees for n.

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