A052886 -id:A052886 - cp's OEIS frontend

A293379 G.f.: exp( Sum_{n>=1} A052886(n) * x^n/n ).

1, 1, 2, 8, 60, 708, 11508, 238356, 5997924, 177615460, 6050014228, 233010132788, 10011985592516, 474815000195844, 24635182609510260, 1388071373846060244, 84406416179170495524, 5509502700599270439588, 384233531128293605883092, 28513258202704586092168820, 2243347226931600749380032388, 186528883284172684793892938308, 16343425532977997699916585695796

Offset: 0

Paul D. Hanna, Oct 22 2017

nonn

The e.g.f. of A052886 equals Series_Reversion( log(1 + x - x^2) ), which also equals (1 - sqrt(5 - 4*exp(x)) )/2.
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 + 8*x^3 + 60*x^4 + 708*x^5 + 11508*x^6 + 238356*x^7 + 5997924*x^8 + 177615460*x^9 + 6050014228*x^10 +...
such that
log(A(x)) = x + 3*x^2/2 + 19*x^3/3 + 207*x^4/4 + 3211*x^5/5 + 64383*x^6/6 + 1581259*x^7/7 + 45948927*x^8/8 + 1541641771*x^9/9 +...+ A052886(n)*x^n/n +...
Note that the e.g.f. of A052886 equals Series_Reversion( log(1 + x - x^2) ) = x + 3*x^2/2! + 19*x^3/3! + 207*x^4/4! + 3211*x^5/5! + 64383*x^6/6! + 1581259*x^7/7! + 45948927*x^8/8! +...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A000108, A052886, A294115.

nmax = 20; A052886 = CoefficientList[Series[(1 - Sqrt[5 - 4*E^x])/2, {x, 0, nmax}], x] * Range[0, nmax]!; Clear[a]; a[n_]:= a[n] = Sum[A052886[[n-k+1]]*a[k], {k, 0, n-1}] / n; a[0] = 1; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 26 2017 *)

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

c(n) = (1/(n+1))*binomial(2*n,n); /* A000108 */
lista(nn)={my(a=vector(nn)); a[1]=1; for(n=2, nn, a[n] = (1/(n-1))*sum(m=1, n-1, m!*c(m-1)*sum(s=m,n-1, stirling(s,m,2)*a[n-s]))); a;} \\ Petros Hadjicostas, Jun 12 2020

a(n) ~ sqrt(5) * n^(n-2) / (2^(3/2) * exp(n) * (log(5/4))^(n-1/2)). - Vaclav Kotesovec, Oct 26 2017

a(n) = (1/n)*Sum_{m=1..n} m! * C(m-1) * Sum_{s=m..n} Stirling2(s,m) * a(n-s) for n >= 1 with a(0) = 1, where C(n) = A000108(n). - Petros Hadjicostas, Jun 12 2020

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

A054687 a(n+1) = 1 + Sum_{k=0..n} binomial(n,k)a(k)a(n-k) for n >= 0 with a(0) = 1.

Original entry on oeis.org

1, 2, 5, 19, 99, 653, 5187, 48117, 510271, 6088301, 80716427, 1177130893, 18727404639, 322769897573, 5990916997611, 119139798166245, 2527255556219167, 56960055683893853, 1359299747696197931, 34240584053654816797, 907911436336049691519, 25277557586594907583733, 737276033151104902965963

Offset: 0

Leroy Quet, Apr 19 2000

			a(5) = 1 + 1*a(0)*a(4) + 4*a(1)*a(3) + 6*a(2)*a(2) + 4*a(3)*a(1) + 1*a(4)*a(0) = 1 + 1*1*99 + 4*2*19 + 6*5*5 + 4*19*2 + 1*99*1 = 653.

Seiichi Manyama, Table of n, a(n) for n = 0..429

Cf. A052886.

nmax=20; b = ConstantArray[0,nmax+2]; b[[1]]=1; Do[b[[n+2]] = 1 + Sum[Binomial[n,k]*b[[k+1]]*b[[n-k+1]],{k,0,n}],{n,0,nmax}]; b (* Vaclav Kotesovec, Mar 02 2014 *)

lista(nn)={my(a=vector(nn)); a[1]=1; for(n=2, nn, a[n]= 1 + sum(k=0, n-2, binomial(n-2, k)*a[k+1]*a[n-k-1])); for(n=1, nn, print1(a[n], ","))}; \\ Petros Hadjicostas, Jun 11 2020

a(n) ~ n! / r^(n+1), where r = 0.7542714558461742549095127982603266798910769877... is the smallest positive real root of the equation (BesselJ(1, 2) - BesselJ(0, 2))*BesselY(0, 2*exp(r/2)) + BesselJ(0, 2*exp(r/2))*(BesselY(0, 2) - BesselY(1, 2)) = 0. - Vaclav Kotesovec, Mar 02 2014, updated Apr 26 2020
E.g.f. A(x) satisfies A'(x) = exp(x) + A(x)^2 with A(0) = 1. - Petros Hadjicostas, Apr 25 2020
E.g.f.: exp(x/2)*(BesselJ(2, 2)*BesselY(1, 2*exp(x/2)) - BesselJ(1, 2*exp(x/2)) * BesselY(2, 2)) / (BesselJ(2, 2)*BesselY(0, 2*exp(x/2)) - BesselJ(0, 2*exp(x/2)) * BesselY(2, 2)). - Vaclav Kotesovec, Apr 26 2020

More terms from James Sellers, Apr 20 2000

A052851 Expansion of e.g.f. 1/2 - (1/2)(1+4log(1-x))^(1/2).

Original entry on oeis.org

0, 1, 3, 20, 220, 3424, 69008, 1706256, 49956240, 1689497376, 64799254752, 2778906776832, 131756614920192, 6843405231815424, 386414606189283072, 23567401521343170048, 1543994621969805135360, 108137637714495023354880, 8062825821198926369725440

Offset: 0

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

Previous name was: A simple grammar.

Seiichi Manyama, Table of n, a(n) for n = 0..360
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 819
Index entries for reversions of series

Cf. A048287, A052886, A087138.
Cf. A371314, A371315.
Cf. A371327, A376041, A376042.

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

CoefficientList[Series[1/2-1/2*(1+4*Log[1-x])^(1/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

a(n):=sum(stirling1(n,k)*k!*binomial(2*k-2,k-1)/k*(-1)^(n+k), k,1,n); /* Vladimir Kruchinin, May 12 2012 */

E.g.f.: 1/2 - (1/2)*(1-4*log(-1/(-1+x)))^(1/2).
a(n) = Sum_{k=1..n} Stirling1(n,k)*k!*C(2*k-2,k-1)/k*(-1)^(n+k). - Vladimir Kruchinin, May 12 2012
a(n) ~ n^(n-1)/(sqrt(2)*exp(3*n/4)*(exp(1/4)-1)^(n-1/2)). - Vaclav Kotesovec, Sep 30 2013
From Seiichi Manyama, Sep 09 2024: (Start)
E.g.f. satisfies A(x) = (-log(1 - x)) / (1 - A(x)).
E.g.f.: Series_Reversion( 1 - exp(-x * (1 - x)) ). (End)

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

A087138 Expansion of (1-sqrt(1-4*log(1+x)))/2.

Original entry on oeis.org

1, 1, 8, 64, 824, 12968, 252720, 5789712, 153169440, 4589004192, 153643615872, 5684390364288, 230307823878144, 10141452865049088, 482259966649655808, 24630247225278881280, 1344614199041549399040, 78137673004382654223360

Offset: 1

Vladeta Jovovic, Oct 18 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..372
Index entries for reversions of series

Cf. A000108, A052851, A052803, A052886, A052895, A006531, A048287, A293604.

Cf. A370462, A370463.

Rest[CoefficientList[Series[(1-Sqrt[1-4*Log[1+x]])/2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, May 03 2015 *)

x='x+O('x^50); Vec(serlaplace((1-sqrt(1-4*log(1+x)))/2)) \\ G. C. Greubel, May 24 2017

a(n) = Sum_{k=1..n} Stirling1(n, k)*k!*Catalan(k-1).
a(n) ~ n! / (2*exp(1/8)*sqrt(Pi) * (exp(1/4)-1)^(n-1/2) * n^(3/2)). - Vaclav Kotesovec, May 03 2015
From Seiichi Manyama, Sep 09 2024: (Start)
E.g.f. satisfies A(x) = (log(1 + x)) / (1 - A(x)).
E.g.f.: Series_Reversion( exp(x * (1 - x)) - 1 ). (End)

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).

A371329 E.g.f. satisfies A(x) = (exp(x/(1 - A(x))) - 1)/(1 - A(x)).

Original entry on oeis.org

0, 1, 5, 58, 1099, 28966, 978669, 40349478, 1964141687, 110251617526, 7010830858753, 498111156585670, 39106669556183475, 3362091299430435846, 314139422902048625717, 31696638229827506705254, 3434797595698979061279727, 397852853779288923308578966

Offset: 0

Seiichi Manyama, Mar 19 2024

nonn

Index entries for reversions of series

Cf. A052892, A371330.

Cf. A052886, A376036, A376037.

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

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

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

A087152 Expansion of (1-sqrt(1-4*log(1+x)))/log(1+x)/2.

Original entry on oeis.org

1, 3, 20, 194, 2554, 42226, 843744, 19769256, 531768120, 16152296424, 546895099200, 20425461026736, 834215500905552, 36988602430554576, 1769524998544143360, 90851799797294235264, 4982968503277896871296

Offset: 1

Vladeta Jovovic, Oct 18 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..370

Cf. A000108, A052851, A052803, A052886, A052895, A006531, A048287.

Rest[CoefficientList[Series[(1-Sqrt[1-4*Log[1+x]])/Log[1+x]/2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, May 03 2015 *)

x='x+O('x^50); Vec(serlaplace((1-sqrt(1-4*log(1+x)))/log(1+x)/2 - 1)) \\ G. C. Greubel, May 24 2017

a(n) = Sum_{k=0..n} Stirling1(n, k)*k!*Catalan(k).

a(n) ~ 2*n! / (exp(1/8)*sqrt(Pi) * (exp(1/4)-1)^(n-1/2) * n^(3/2)). - Vaclav Kotesovec, May 03 2015

A117269 Triangle T, read by rows, that satisfies matrix equation: T - (T-I)^2 = C, where C is Pascal's triangle.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 19, 9, 3, 1, 207, 76, 18, 4, 1, 3211, 1035, 190, 30, 5, 1, 64383, 19266, 3105, 380, 45, 6, 1, 1581259, 450681, 67431, 7245, 665, 63, 7, 1, 45948927, 12650072, 1802724, 179816, 14490, 1064, 84, 8, 1, 1541641771, 413540343, 56925324, 5408172

Offset: 0

Paul D. Hanna, Mar 05 2006

E.g.f. of column 0 is F(x) = (3-sqrt(5-4*exp(x)))/2 since F(x) satisfies the characteristic equation: F - (F-1)^2 = exp(x). The matrix log of T is the integer triangle A117270.

			Triangle T begins:
1;
1,1;
3,2,1;
19,9,3,1;
207,76,18,4,1;
3211,1035,190,30,5,1;
64383,19266,3105,380,45,6,1;
1581259,450681,67431,7245,665,63,7,1; ...
where (T-I)^2 =
0;
0,0;
2,0,0;
18,6,0,0;
206,72,12,0,0;
3210,1030,180,20,0,0;
64382,19260,3090,360,30,0,0; ...
and T - (T-I)^2 = Pascal's triangle.

Cf. A117270 (log), A117271, A052886.

{T(n,k)=local(C=matrix(n+1,n+1,r,c,if(r>=c,binomial(r-1,c-1))),M=C); for(i=1,n+1,M=(M-M^0)^2+C);return(M[n+1,k+1])}

T(n,k) = A052886(n-k)*C(n,k) for n>k, with T(n,n) = 1.

cp's OEIS Frontend

A293379 G.f.: exp( Sum_{n>=1} A052886(n) * x^n/n ).

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A277120 Number of branching factorizations of n.

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C

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A054687 a(n+1) = 1 + Sum_{k=0..n} binomial(n,k)*a(k)*a(n-k) for n >= 0 with a(0) = 1.

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A052851 Expansion of e.g.f. 1/2 - (1/2)*(1+4*log(1-x))^(1/2).

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Maple

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Maxima

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A087138 Expansion of (1-sqrt(1-4*log(1+x)))/2.

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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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A054687 a(n+1) = 1 + Sum_{k=0..n} binomial(n,k)a(k)a(n-k) for n >= 0 with a(0) = 1.

A052851 Expansion of e.g.f. 1/2 - (1/2)(1+4log(1-x))^(1/2).