A000311 -id:A000311 - cp's OEIS frontend

A201465 E.g.f. satisfies: A(x) = (x + 2*exp(A(x)) - 2)/3.

1, 2, 14, 162, 2622, 54546, 1386702, 41660226, 1444071006, 56728401138, 2490626473326, 120858220146978, 6423145784929278, 371046277074303954, 23148851187463826958, 1551182540888019542274, 111111330526583477368734, 8472364399282482984295602, 685178683361064789536947374

Offset: 1

Paul D. Hanna, Dec 01 2011

nonn

			E.g.f.: A(x) = x + 2*x^2 + 14*x^3/3! + 162*x^4/4! + 2622*x^5/5! + 54546*x^6/6! +...
The exponential of the e.g.f. begins:
exp(A(x)) = 1 + x + 3*x^2/2! + 21*x^3/3! + 243*x^4/4! + 3933*x^5/5! + 81819*x^6/6! +...
where x = 2 + 3*A(x) - 2*exp(A(x)).
...
O.g.f.: G(x) = 1 + 2*x + 14*x^2 + 162*x^3 + 2622*x^4 + 54546*x^5 +...
where
G(x) = 1/3 + 2/(3*(3-x)) + 2^2/(3*(3-x)*(3-2*x)) + 2^3/(3*(3-x)*(3-2*x)*(3-3*x)) + 2^4/(3*(3-x)*(3-2*x)*(3-3*x)*(3-4*x)) + 2^5/(3*(3-x)*(3-2*x)*(3-3*x)*(3-4*x)*(3-5*x)) +...

Cf. variants: A000311, A201466. A058562.

Rest[CoefficientList[1 + InverseSeries[Series[2 + 3*x - 2*Exp[x], {x, 0, 20}], x],x]* Range[0, 20]!] (* Vaclav Kotesovec, Dec 26 2013 *)

a(n):=(sum((n+k-1)!*sum(1/(k-j)!*sum((3^i*(-1)^(i)*2^(j-i)*stirling2(n+j-i-1,j-i))/(i!*(n+j-i-1)!),i,0,j),j,0,k),k,0,n-1)); /* Vladimir Kruchinin, Feb 04 2012 */

{a(n)=n!*polcoeff(serreverse(2+3*x - 2*exp(x+x^2*O(x^n))),n)}
for(n=0, 25, print1(round(A[n+1]), ", "))

\p100 \\ set precision
{A=Vec(sum(n=0, 600, 1.*2^n/prod(k=0, n, 3 - k*x + O(x^31))))}
for(n=0, 25, print1(round(A[n+1]), ", ")) \\ Paul D. Hanna, Oct 27 2014

E.g.f. A(x) satisfies: x = A( 2 + 3*x - 2*exp(x) ).
a(n)=(sum(k=0..n-1, (n+k-1)!*sum(j=0..k, 1/(k-j)!*sum(i=0..j, (3^i*(-1)^(i)*2^(j-i)*stirling2(n+j-i-1,j-i))/(i!*(n+j-i-1)!))))), n>0. [From Vladimir Kruchinin, Feb 04 2012]
exp(A(x))-1 is the compositional inverse of 3*log(1+x)-2*x and is the e.g.f. of A058562. - Peter Bala, Jul 12 2012
G.f.: 1/Q(0), where Q(k)= 1 - k*x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
E.g.f.: (x-2)/3 - LambertW(-2/3*exp((x-2)/3)). - Vaclav Kotesovec, Dec 26 2013
a(n) ~ n^(n-1) / (sqrt(3) * exp(n) * (3*log(3)-3*log(2)-1)^(n-1/2)). - Vaclav Kotesovec, Dec 26 2013
O.g.f.: Sum_{n>=0} 2^n / Product_{k=0..n} (3 - k*x). - Paul D. Hanna, Oct 27 2014

A268163 Number of labeled binary-ternary rooted non-planar trees, indexed by number of leaves.

Original entry on oeis.org

0, 1, 1, 4, 25, 220, 2485, 34300, 559405, 10525900, 224449225, 5348843500, 140880765025, 4063875715900, 127418482316125, 4314607214417500, 156920190449147125, 6100643259005795500, 252476539015516440625, 11081983532721088487500, 514215436341672155715625

Offset: 0

Murray R. Bremner, Jan 27 2016

nonn

This can also be interpreted as the number of multilinear monomials of degree n in a nonassociative algebra with an (anti)commutative binary operation and a completely (skew-)symmetric ternary operation; the number of variables in the monomial corresponds to the number of leaves in the tree.

This sequence also enumerates a certain class of Feynman diagrams; see the references, links, and crossrefs below.

			For n = 4 and using the monomial interpretation, the 25 multilinear monomials of degree 4 are as follows, where [-,-] is the binary operation and (-,-,-) is the ternary operation:
[[[a,b],c],d], [[[a,b],d],c], [[[a,c],b],d], [[[a,c],d],b], [[[a,d],b],c], [[[a,d],c],b], [[[b,c],a],d], [[[b,c],d],a], [[[b,d],a],c], [[[b,d],c],a], [[[c,d],a],b], [[[c,d],b],a], [[a,b],[c,d]], [[a,c],[b,d]], [[a,d],[b,c]], [(a,b,c),d], [(a,b,d),c], [(a,c,d),b], [(b,c,d),a], ([a,b],c,d), ([a,c],b,d), ([a,d],b,c), ([b,c],a,d), ([b,d],a,c), ([c,d],a,b).

J. Bedford, On Perturbative Field Theory and Twistor String Theory, Ph.D. Thesis, 2007, Queen Mary, University of London.
B. Feng and M. Luo, An introduction to on-shell recursion relations, Review Article, Frontiers of Physics, October 2012, Volume 7, Issue 5, pp. 533-575.
K. Kampf, A new look at the nonlinear sigma model, 17th International Conference in Quantum Chromodynamics (QCD 14), Nuclear and Particle Physics Proceedings, Volumes 258-259, January-February 2015, pp. 86-89.
M. L. Mangano and S. J. Parke, Multi-parton amplitudes in gauge theories, Physics Reports, Volume 200, Issue 6, February 1991, pp. 301-367.

Alois P. Heinz, Table of n, a(n) for n = 0..400.
J. Bedford, On Perturbative Field Theory and Twistor String Theory, arXiv:0709.3478 [hep-th], 2007, (see page 23).
D. Carmi, TeV scale strings and scattering amplitudes at the LHC, arXiv:1109.5161 [hep-th], 2011-2015, (see page 23).
B. Feng, M. Luo, An introduction to on-shell recursion relations, arXiv:1111.5759 [hep-th], 2011-2012, (see page 2).
K. Kampf, A new look at the nonlinear sigma model (see pages 7 and 13).
M. L. Mangano, S. J. Parke, Multi-parton amplitudes in gauge theories, arXiv:hep-th/0509223, 2005, (see page 5).
G. Travaglini, Harmony of (super) form factors (see page 3).
A. Volovich, Yang-Mills amplitudes and twistor string theory (see bottom of page 1).
James Christopher Whitehead, The Production of Pairs of Isolated Photons at Higher Orders in QCD, Ph. D. Thesis, Durham University (UK, 2021).
Wikipedia, Feynman diagram

Cf. A001147. The number of labeled binary rooted non-planar trees.
Cf. A025035. The number of labeled ternary rooted non-planar trees.
Cf. A268172. The corresponding number of unlabelled trees.
Cf. A005411. Number of non-vanishing Feynman diagrams of order 2n for the electron or the photon propagators in quantum electrodynamics.
Cf. A005412. Number of non-vanishing Feynman diagrams of order 2n for the vacuum polarization (the proper two-point function of the photon) and for the self-energy (the proper two-point function of the electron) in quantum electrodynamics (QED).
Cf. A005413. Number of non-vanishing Feynman diagrams of order 2n+1 for the electron-electron-photon proper vertex function in quantum electrodynamics (QED).
Cf. A005414. Feynman diagrams of order 2n with vertex skeletons.
Other sequences related to Feynman diagrams: A115974, A122023, A167872, A214298, A214299.
Cf. A000311.

with(combinat):
b:= proc(n, i, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or nAlois P. Heinz, Jan 28 2016
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [0, 1$2][n+1],
       ((24*n-36)*a(n-1)+(3*n-5)*(3*n-7)*a(n-2))/11)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 28 2016

a[0]=0; a[1]=1; a[2]=1; a[n_]:=a[n]=(12(2n-3)a[n-1]+(3n-5)(3n-7)a[n-2])/11; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)

a(n) = ((24*n-36)*a(n-1)+(3*n-5)*(3*n-7)*a(n-2))/11 for n>2. - Alois P. Heinz, Jan 28 2016
Because of Koszul duality for operads, the exponential generating function is the compositional inverse of the power series x-x^2/2-x^3/6 (email of Vladimir Dotsenko to Murray R. Bremner, Jan 28 2016).
a(n) ~ sqrt(9-4*sqrt(3)) * ((12+9*sqrt(3))/11)^n * n^(n-1) / (3 * exp(n)). - Vaclav Kotesovec, Feb 24 2016

A300402 Smallest integer i such that TREE(i) >= n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Felix Fröhlich, Mar 05 2018

nonn

The sequence grows very slowly.
A rooted tree is a tree containing one special node labeled the "root".
TREE(n) gives the largest integer k such that a sequence T(1), T(2), ..., T(k) of vertex-colored (using up to n colors) rooted trees, each one T(i) having at most i vertices, exists such that T(i) <= T(j) does not hold for any i < j <= k. - Edited by Gus Wiseman, Jul 06 2020

			TREE(1) = 1, so a(n) = 1 for n <= 1.
TREE(2) = 3, so a(n) = 2 for 2 <= n <= 3.
TREE(3) > A(A(...A(1)...)), where A(x) = 2[x+1]x is a variant of Ackermann's function, a[n]b denotes a hyperoperation and the number of nested A() functions is 187196, so a(n) = 3 for at least 4 <= n <= A^A(187196)(1).

Priyabrata Biswas, Towards Data Science: How Big Is The Number — Tree(3)
Eric Weisstein's World of Mathematics, Rooted Tree
Wikipedia, Hyperoperation - Notations
Wikipedia, Kruskal's tree theorem

Cf. A090529, A300403, A300404.
Labeled rooted trees are counted by A000169 and A206429.
Cf. A000081, A000311, A060313, A060356, A317713.

A318577 Number of complete multimin tree-factorizations of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 11, 1, 3, 1, 4, 1, 1, 1, 19, 1, 1, 3, 4, 1, 4, 1, 45, 1, 1, 1, 17, 1, 1, 1, 19, 1, 4, 1, 4, 4, 1, 1, 96, 1, 3, 1, 4, 1, 11, 1, 19, 1, 1, 1, 26, 1, 1, 4, 197, 1, 4, 1, 4, 1, 4, 1, 104, 1, 1, 3, 4, 1, 4, 1, 96, 11, 1, 1, 26, 1, 1, 1, 19, 1, 19, 1, 4, 1, 1, 1, 501, 1, 3, 4, 17

Offset: 1

Gus Wiseman, Sep 11 2018

nonn

A multimin factorization of n is an ordered factorization of n into factors greater than 1 such that the sequence of minimal primes dividing each factor is weakly increasing. A multimin tree-factorization of n is either the number n itself or a sequence of at least two multimin tree-factorizations, one of each factor in a multimin factorization of n. A multimin tree-factorization is complete if the leaves are all prime numbers.

			The a(12) = 4 trees are (2*2*3), (2*(2*3)), ((2*3)*2), ((2*2)*3).

Cf. A000311, A001003, A001055, A020639, A255397, A281113, A281118, A281119, A295281, A317545, A319118, A319121.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];
Table[Length[Select[mmftrees[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,100}]

a(prime^n) = A001003(n - 1).

a(product of n distinct primes) = A000311(n).

A058476 Total number of multiple edges in all essentially parallel series-parallel networks with n labeled edges, multiple edges allowed.

Original entry on oeis.org

0, 0, 1, 2, 21, 224, 3075, 50364, 958881, 20786744, 505233987, 13603353928, 401819713273, 12917450788956, 448922610588491, 16770152528046332, 670096317408222529, 28517474585339343408, 1287722135213121579203

Offset: 0

N. J. A. Sloane, Dec 20 2000

nonn

J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence M_P(n)*P_pi).

Index entries for sequences mentioned in Moon (1987)

E.g.f. = (exp(-x)-1+x)*P'(x), where P(x) = e.g.f. for A000311.

A200318 E.g.f. satisfies: A(x) = x-1 + cosh(A(x)).

Original entry on oeis.org

1, 1, 3, 16, 120, 1156, 13608, 189316, 3039060, 55291336, 1124309208, 25268818576, 622008616320, 16642670404816, 480923246983728, 14926731083999296, 495243684302520000, 17491488288340789696, 655224017429959987968, 25947019896579324410176, 1083050878686674070676800

Offset: 1

Paul D. Hanna, Nov 15 2011

nonn

a(n) is the number of leaf labeled rooted trees with n leaves in which the outdegrees of the root and all internal nodes are positive even integers. - Geoffrey Critzer, Jul 31 2016

			E.g.f.: A(x) = x + x^2/2! + 3*x^3/3! + 16*x^4/4! + 120*x^5/5! +...
where A(1+x - cosh(x)) = x and A(x) = x-1 + cosh(A(x)).
The e.g.f. satisfies:
A(x) = x + (cosh(x)-1) + d/dx (cosh(x)-1)^2/2! + d^2/dx^2 (cosh(x)-1)^3/3! + d^3/dx^3 (cosh(x)-1)^4/4! +...
as well as the logarithmic series:
log(A(x)/x) = (cosh(x)-1)/x + d/dx (cosh(x)-1)^2/x/2! - d^2/dx^2 (cosh(x)-1)^3/x/3! + d^3/dx^3 (cosh(x)-1)^4/x/4! +...

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 128,(labeled hierarchies).

Cf. A200317, A000311, A052526.

Rest[CoefficientList[InverseSeries[Series[1 + x - Cosh[x],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 10 2014 *)

{a(n)=n!*polcoeff(serreverse(1+x-cosh(x+x^2*O(x^n))),n)}
for(n=1, 21, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, (cosh(x+x*O(x^n))-1)^m)/m!)+x*O(x^n)); n!*polcoeff(A, n)}

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, (cosh(x+x*O(x^n))-1)^m/x)/m!)+x*O(x^n))); n!*polcoeff(A, n)}

E.g.f. satisfies:
(1) A(x) = Series_Reversion(1+x - cosh(x)).
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (cosh(x) - 1)^n / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (cosh(x) - 1)^n/x / n! ).
a(n) ~ n^(n-1) / (2^(1/4) * exp(n) * (1-sqrt(2)+log(1+sqrt(2)))^(n-1/2)). - Vaclav Kotesovec, Jan 10 2014

A201466 E.g.f. satisfies: A(x) = (x + 3*exp(A(x)) - 3)/4.

Original entry on oeis.org

1, 3, 30, 498, 11568, 345432, 12606240, 543678672, 27054328512, 1525746223488, 96167433279360, 6699404849841408, 511152613463843328, 42391161255859802112, 3796840836492517125120, 365260399012767192102912, 37561729737177160757133312, 4111876748834828077514170368

Offset: 1

Paul D. Hanna, Dec 01 2011

nonn

			E.g.f.: A(x) = x + 3*x^2/2! + 30*x^3/3! + 498*x^4/4! + 11568*x^5/5! + 345432*x^6/6! +...
The exponential of the e.g.f. begins:
exp(A(x)) = 1 + x + 4*x^2/2! + 40*x^3/3! + 664*x^4/4! + 15424*x^5/5! + 460576*x^6/6! +...
where x = 3 + 4*A(x) - 3*exp(A(x)).
...
O.g.f.: G(x) = 1 + 3*x + 30*x^2 + 498*x^3 + 11568*x^4 + 345432*x^5 +...
where
G(x) = 1/4 + 3/(4*(4-x)) + 3^2/(4*(4-x)*(4-2*x)) + 3^3/(4*(4-x)*(4-2*x)*(4-3*x)) + 3^4/(4*(4-x)*(4-2*x)*(4-3*x)*(4-4*x)) + 3^5/(4*(4-x)*(4-2*x)*(4-3*x)*(4-4*x)*(4-5*x)) +...

Cf. variants: A000311, A201465.

Rest[CoefficientList[InverseSeries[Series[3 - 3*E^x + 4*x,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 10 2014 *)

{a(n)=n!*polcoeff(serreverse(3+4*x - 3*exp(x+x^2*O(x^n))),n)}

\p100 \\ set precision
{A=Vec(sum(n=0, 600, 1.*3^n/prod(k=0, n, 4 - k*x + O(x^31))))}
for(n=0, 25, print1(round(A[n+1]), ", ")) \\ Paul D. Hanna, Oct 27 2014

E.g.f. satisfies: x = A( 3 + 4*x - 3*exp(x) ).
E.g.f.: (x-3)/4 - LambertW(-3*exp((x-3)/4)/4). - Vaclav Kotesovec, Jan 10 2014
a(n) ~ n^(n-1) / (2 * (4*log(4/3)-1)^(n-1/2) * exp(n)). - Vaclav Kotesovec, Jan 10 2014
O.g.f.: Sum_{n>=0} 3^n / Product_{k=0..n} (4 - k*x). - Paul D. Hanna, Oct 27 2014

A298673 Inverse matrix of A135494.

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 26, 19, 6, 1, 236, 170, 55, 10, 1, 2752, 1966, 645, 125, 15, 1, 39208, 27860, 9226, 1855, 245, 21, 1, 660032, 467244, 155764, 32081, 4480, 434, 28, 1, 12818912, 9049584, 3031876, 635124, 92001, 9576, 714, 36, 1, 282137824, 198754016, 66845340, 14180440, 2108085, 230097, 18690, 1110, 45, 1

Offset: 1

Tom Copeland, Jan 24 2018

Since this is the inverse matrix of A135494 with row polynomials q_n(t), first introduced in that entry by R. J. Mathar, and the row polynomials p_n(t) of this entry are a binomial Sheffer polynomial sequence, the row polynomials of the inverse pair are umbral compositional inverses, i.e., p_n(q.(t)) = q_n(p.(t)) = t^n. For example, p_3(q.(t)) = 4q_1(t) + 3q_2(t) + q_3(t) = 4t + 3(-t + t^2) + (-t -3t^2 +t^3) = t^3. In addition, both sequences possess the umbral convolution property (p.x) + p.(y))^n = p_n(x+y) with p_0(t) = 1.

This is the inverse of the Bell matrix generated by A153881; for the definition of the Bell matrix see the link. - Peter Luschny, Jan 26 2018

			Matrix begins as
     1;
     1;    1;
     4,    3,    1;
    26,   19,    6,    1;
   236,  170,   55,   10,    1;
  2752, 1966,  645,  125,   15,    1;

Peter Luschny, The Bell transform
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.

Cf. A000311, A000169, A000670, A006351, A135494.

# The function BellMatrix is defined in A264428. Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n=0, 1, -1), 9): MatrixInverse(%); # Peter Luschny, Jan 26 2018

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, If[n == 0, 1, -1]], rows = 12] // Inverse;
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

E.g.f.: e^[p.(t)x] = e^[t*h(x)] = exp[t*[(x-1)/2 + T{ (1/2) * exp[(x-1)/2] }], where T is the tree function of A000169 related to the Lambert function. h(x) = sum(j=1,...) A000311(j) * x^j / j! = exp[xp.'(0)], so the first column of this entry's matrix is A000311(n) for n > 0 and the second column of the full matrix for p_n(t) to n >= 0. The compositional inverse of h(x) is h^(-1)(x) = 1 + 2x - e^x.
The lowering operator is L = h^(-1)(D) = 1 + 2D - e^D with D = d/dt, i.e., L p_n(t) = n * p_(n-1)(t). For example, L p_3(t) = (D - D^2! - D^3/3! - ...) (4t + 6t^ + t^3) = 3 (t + t^2) = 3 p_2(t).
The raising operator is R = t * 1/[d[h^(-1)(D)]/dD] = t * 1/[2 - e^D)] = t (1 + D + 3D^2/2! + 13D^3/3! + ...). The coefficients of R are A000670. For example, R p_2(t) = t (1 + D + 3D^2/2!  + ...)  (t + t^2) = 4t + 3t^2 + t^3   = p_3(t).
The row sums are A006351, or essentially 2*A000311.
Conjectures from Mikhail Kurkov, Mar 01 2025: (Start)
T(n,k) = Sum_{j=0..n-k} binomial(n+j-1, k-1)*A269939(n-k, j) for 1 <= k <= n.
T(n,k) = A(n-1,k,0) for n > 0, k > 0 where A(n,k,q) = A(n-1,k,q+1) + 2*(q+1)!*Sum_{j=0..q} A(n-1,k,j)/j! for n >= 0, k > 0, q >= 0 with A(0,k,q) = Stirling1(q+1,k) for k > 0, q >= 0 (see A379458). In other words, T(n,k) = Sum_{j=0}^{n-1} A379460(n-j-1,j)*Stirling1(j+1,k) for n > 0, k > 0.
Recursion for the n-th row (independently of other rows): T(n,k) = 1/(n-k)*Sum_{j=2..n-k+1} b(j-1)*binomial(-k,j)*T(n,k+j-1)*(-1)^j for 1 <= k < n with T(n,n) = 1 where b(n) = 1 + 4*Sum_{i=1..n} A135148(i).
Recursion for the k-th column (independently of other columns): T(n,k) = 1/(n-k)*Sum_{j=2..n-k+1} c(j-1)*binomial(n,j)*T(n-j+1,k) for 1 <= k < n with T(n,n) = 1 where c(n) = A000311(n+1) + (n-1)*A000311(n). (End)

A306576 Expansion of 1/(1 - x - 2x/(1 - 2x - 3x/(1 - 3x - 4x/(1 - 4x - 5*x/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 3, 19, 179, 2183, 32355, 562343, 11198203, 251297263, 6275390067, 172639089031, 5189033793611, 169220733646271, 5951777459480931, 224604052936701815, 9053124776482735291, 388198017158108201839, 17645733672934447166163, 847577245047341210277415

Offset: 0

Ilya Gutkovskiy, Jun 25 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..386

Cf. A000311, A001147, A003319, A158882.

$RecursionLimit = Infinity; nmax = 18; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-k x, 1 - k x, {k, 2, nmax + 1}]), {x, 0, nmax}], x]

a(n) ~ c * d^n * n^(n+1), where d = 1 / (exp(1) * (2*log(2) - 1)) = 0.952329306865721945... and c = 1/(sqrt(2) * (2*log(2) - 1)^(3/2)) = 2.945150206105358... - Vaclav Kotesovec, Jul 01 2019, updated May 06 2024

A318120 Number of set partitions of {1,...,n} with relatively prime block sizes.

Original entry on oeis.org

1, 1, 1, 4, 11, 51, 162, 876, 3761, 20782, 109293, 678569, 4038388, 27644436, 186524145, 1379760895, 10323844183, 82864869803, 674798169662, 5832742205056, 51385856585637, 474708148273586, 4486977535287371, 44152005855084345, 444577220573083896

Offset: 0

Gus Wiseman, Dec 16 2018

nonn

			The a(4) = 11 set partitions:
  {{1},{2},{3},{4}}
   {{1},{2},{3,4}}
   {{1},{2,3},{4}}
   {{1},{2,4},{3}}
   {{1,2},{3},{4}}
   {{1,3},{2},{4}}
   {{1,4},{2},{3}}
    {{1},{2,3,4}}
    {{1,2,3},{4}}
    {{1,2,4},{3}}
    {{1,3,4},{2}}

Alois P. Heinz, Table of n, a(n) for n = 0..576

a(n) >= A271426(n).

Cf. A000110, A000258, A000311, A000670, A000740, A000837, A005651, A005804, A008277, A124794, A319182, A320289.

b:= proc(n, t) option remember; `if`(n=0, `if`(t<2, 1, 0),
      add(b(n-j, igcd(t, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 30 2019

numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[Total[numSetPtnsOfType/@Select[IntegerPartitions[n],GCD@@#==1&]],{n,10}]
(* Second program: *)
b[n_, t_] := b[n, t] = If[n == 0, If[t < 2, 1, 0],
     Sum[b[n - j, GCD[t, j]]*Binomial[n - 1, j - 1], {j, 1, n}]];
a[n_] := b[n, 0];
a /@ Range[0, 25] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(n) = Sum_{|y| = n, GCD(y) = 1} n! / (Product_i y_i! * Product_i (y)_i!) where the sum is over all relatively prime integer partitions of n and (y)_i is the multiplicity of i in y.

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