A034940 -id:A034940 - cp's OEIS frontend

A034941 Number of labeled triangular cacti with 2n+1 nodes (n triangles).

1, 1, 15, 735, 76545, 13835745, 3859590735, 1539272109375, 831766748637825, 585243816844111425, 520038240188935042575, 569585968715180280038175, 753960950911045074462890625, 1186626209895384011075327630625, 2190213762744801162239116550679375

Offset: 0

Christian G. Bower, Oct 15 1998

nonn

Also the number of 3-uniform hypertrees spanning 2n + 1 labeled vertices. - Gus Wiseman, Jan 12 2019

Number of rank n+1 simple series-parallel matroids on [2n+1]. - Matt Larson, Mar 06 2023

			a(3) = 5!! * 7^2 = (1*3*5) * 49 = 735.
From _Gus Wiseman_, Jan 12 2019: (Start)
The a(2) = 15 3-uniform hypertrees:
  {{1,2,3},{1,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{3,4,5}}
  {{1,2,5},{1,3,4}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,3,4},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,4,5},{2,3,4}}
  {{1,4,5},{2,3,5}}
The following are non-isomorphic representatives of the 2 unlabeled 3-uniform hypertrees spanning 7 vertices, and their multiplicities in the labeled case, which add up to a(3) = 735:
  105 X {{1,2,7},{3,4,7},{5,6,7}}
  630 X {{1,2,6},{3,4,7},{5,6,7}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..200
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
Luis Ferroni and Matt Larson, Kazhdan-Lusztig polynomials of braid matroids, arXiv:2303.02253 [math.CO], 2023.
Katie Gedeon, N. Proudfoot, and B. Young, Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures, arXiv preprint arXiv:1611.07474 [math.CO], 2016-2017.
Nicholas Proudfoot and Ben Young, Configuration spaces, FS^op-modules, and Kazhdan-Lusztig polynomials of braid matroids, arXiv:1704.04510 [math.RT], 2017.
Eric Weisstein's World of Mathematics, Cactus Graph
Index entries for sequences related to cacti

Cf. A000272, A000665, A003081, A030019, A035053, A125791, A302374, A320444, A323292, A323298.

[(2*n+1)^(n-1)*Factorial(2*n)/(2^n*Factorial(n)): n in [0..15]]; // Vincenzo Librandi, Feb 19 2020

Table[(2n+1)^(n-1)(2n)!/(2^n n!), {n, 0, 14}] (* Jean-François Alcover, Nov 06 2018 *)

a(n) = A034940(n)/(2n+1).

The closed form a(n) = (2n-1)!! (2n+1)^(n-1) can be obtained from the generating function in A034940. - Noam D. Elkies, Dec 16 2002

Typo in a(10) corrected and more terms from Alois P. Heinz, Jun 23 2017

A214406 Triangle of second-order Eulerian numbers of type B.

Original entry on oeis.org

1, 1, 1, 1, 8, 3, 1, 33, 71, 15, 1, 112, 718, 744, 105, 1, 353, 5270, 14542, 9129, 945, 1, 1080, 33057, 191384, 300291, 129072, 10395, 1, 3265, 190125, 2033885, 6338915, 6524739, 2071215, 135135, 1, 9824, 1038780, 18990320, 103829590, 204889344, 150895836, 37237680, 2027025

Offset: 0

Peter Bala, Jul 17 2012

The second-order Eulerian numbers A008517 count Stirling permutations by ascents. A Stirling permutation of order n is a permutation of the multiset {1,1,2,2,...,n,n} such that for each i, 1 <= i <= n, the elements lying between the two occurrences of i are larger than i.

We define a signed Stirling permutation of order n to be a vector (x_0, x_1, ..., x_(2*n)) such that x_0 = 0 and (|x_1|, ... ,|x_(2*n)|) is a Stirling permutation of order n. We say that a signed Stirling permutation (x_0, x_1, ... , x_(2*n)) has an ascent at position j, 0 <= j <= 2*n-1, if |x_j| < |x_(j+1)|. We define T(n,k), the second-order Eulerian numbers of type B, as the number of signed Stirling permutations of order n having k ascents. An example is given below.

			Row 2: [1,8,3]:
Signed Stirling permutations of order 2
= = = = = = = = = = = = = = = = = = = =
..............ascents...................ascents
(0 2 2 1 1)......1.......(0 -2 -2 1 1).....1
(0 1 2 2 1)......2.......(0 1 -2 -2 1).....2
(0 1 1 2 2)......2.......(0 1 1 -2 -2).....1
(0 2 2 -1 -1)....1.......(0 -2 -2 -1 -1)...1
(0 -1 2 2 -1)....1.......(0 -1 -2 -2 -1)...1
(0 -1 -1 2 2)....1.......(0 -1 -1 -2 -2)...0
............................................
Triangle begins
.n\k.|..0.....1......2.......3......4........5......6
= = = = = = = = = = = = = = = = = = = = = = = = = = =
..0..|..1
..1..|..1.....1
..2..|..1.....8......3
..3..|..1....33.....71......15
..4..|..1...112....718.....744....105
..5..|..1...353...5270...14542...9129......945
..6..|..1..1080..33057..191384..300291..129072..10395
...
Recurrence example: T(4,2) = 11*T(3,1) + 5*T(3,2) = 11*33 + 5*71 = 718.

Peter Bala, Second-order Eulerian numbers of type B
Peter Bala, Notes on A214406
L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros arXiv:math/0509207 [math.CO], 2005-2006.

Cf. A001813 (row sums), A008517, A039755, A185896, A034940.

T[n_, k_] /; 0 < k <= n := T[n, k] = (4n-2k-1) T[n-1, k-1] + (2k+1) T[n-1, k]; T[, 0] = 1; T[, _] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 11 2019 *)

T(n,k) = Sum_{i = 0..k} (-1)^(i-k)*binomial(2*n+1,k-i)*S(n+i,i), where S(n,k) = 1/(2^k*k!)*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*(2*j+1)^n = A039755(n,k).
It appears that Sum_{k = 0..n} (-1)^(k+1)*T(n,k)/((2*n-k)*binomial(2*n,k)) = (-1)^n *(2^n-2)*Bernoulli(n)/n.
Recurrence equation: T(n,k) = (4*n-2*k-1)*T(n-1,k-1) + (2*k+1)*T(n-1,k), for n,k >= 0.
The row polynomials R(n,x) may be calculated by means of the recurrence equation R(0,x) = 1 and for n >= 0, R(n,x^2) = (1 - x^2)^(2*n)*d/dx( x/(1-x^2)^(2*n-1)*R(n-1,x^2) ). Equivalently, x*R(n,x^2)/(1 - x^2)^(2*n+1)) = D^n(x), where D is the differential operator x/(1 - x^2)*d/dx.
Another recurrence is R(n+1,x) = 2*x*(1 - x)*d/dx(R(n,x)) + (1 + (4*n+1)*x)*R(n,x). It follows that the row polynomials R(n,x) have only real zeros (apply Liu and Wang, Corollary 1.2 with f(x) = R(n,x) and g(x) = R'(n,x)).
For n >= 0, the rational functions Q(n,x) := R(n,x)/(1 - x)^(2*n+1) are the o.g.f.'s for the diagonals of the type B Stirling numbers of the second kind A039755. They appear to satisfy the semi-orthogonality property Integral_{x = 0..oo} (1 - x)*Q(n,x)*Q(m,x) dx = (-1)^n*(2^(n+m) - 2)*Bernoulli(n+m)/(n+m), for n, m >= 0 but excluding the case (n,m) = (0,0). A similar result holds for the row polynomials of A185896.
Row sums are A001813.
Define functions F(n,z) := Sum_{k >= 0} (2*k+1)^(k+n)*z^k/k!, n = 0,1,2,.... Then exp(-x/2)*F(n,x/2*exp(-x)) = R(n,x)/(1 - x)^(2*n+1). - Peter Bala, Jul 26 2012

Missing 1 in data inserted by Jean-François Alcover, Nov 11 2019

A003080 Number of rooted triangular cacti with 2n+1 nodes (n triangles).

Original entry on oeis.org

1, 1, 2, 5, 13, 37, 111, 345, 1105, 3624, 12099, 41000, 140647, 487440, 1704115, 6002600, 21282235, 75890812, 272000538, 979310627, 3540297130, 12845634348, 46764904745, 170767429511, 625314778963, 2295635155206, 8447553316546, 31153444946778, 115122389065883

Offset: 0

N. J. A. Sloane

a(n) is also the number of isomorphism classes of Fano Bott manifolds of complex dimension n (see [Cho-Lee-Masuda-Park]). - Eunjeong Lee, Jun 29 2021

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 305, (4.2.34).
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 73, (3.4.20).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
Yunhyung Cho, Eunjeong Lee, Mikiya Masuda, and Seonjeong Park, On the enumeration of Fano Bott manifolds, arXiv:2106.12788 [math.AG], 2021. See Table 1 p. 8.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1 (1992) pp. 53-80.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
N. J. A. Sloane, Transforms
Index entries for sequences related to cacti

Column k=3 of A332648.

Cf. A003081, A034940, A034941.

terms = 30;
nmax = 2 terms;
A[] = 0; Do[A[x] = x Exp[Sum[(A[x^n]^2 + A[x^(2n)])/(2n), {n, 1, terms}]] + O[x]^nmax // Normal, {nmax}];
DeleteCases[CoefficientList[A[x], x], 0] (* Jean-François Alcover, Sep 02 2018 *)

a(n)=b(2n+1). b shifts left under transform T where Tb = EULER(E_2(b)). E_2(b) has g.f. (B(x^2)+B(x)^2)/2.

a(n) ~ c * d^n / n^(3/2), where d = 3.90053254788870206167147120260433375638561926371844809... and c = 0.4861961460367182791173441493565088408563977498871021... - Vaclav Kotesovec, Jul 01 2021

Sequence extended by Paul Zimmermann, Mar 15 1996

Additional comments from Christian G. Bower

A003081 Number of triangular cacti with 2n+1 nodes (n triangles).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 19, 48, 126, 355, 1037, 3124, 9676, 30604, 98473, 321572, 1063146, 3552563, 11982142, 40746208, 139573646, 481232759, 1669024720, 5819537836, 20390462732, 71762924354, 253601229046, 899586777908, 3202234779826, 11435967528286, 40964243249727

Offset: 0

N. J. A. Sloane

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 306, (4.2.35).
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 73, (3.4.21).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1 (1992) pp. 53-80.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
Index entries for sequences related to cacti

Column k=3 of A332649.

Cf. A003080, A034940, A034941.

terms = 31;
nmax = 2 terms;
A[_] = 0;
Do[A[x_] = x Exp[Sum[(A[x^n]^2 + A[x^(2n)])/(2n), {n, 1, terms}]] + O[x]^nmax // Normal, {nmax}];
g[x_] = (A[x] /. x^k_ -> x^((k - 1)/2)) - x + 1;
g[x] + x((g[x^3] - g[x]^3)/3) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2020, after Andrew Howroyd *)

a(n)=b(2n+1). A003080(n)=c(2n+1).
G.f.: B(x)=C(x)+(C(x^3)-C(x)^3)/3.
G.f.: g(x) + x*(g(x^3) - g(x)^3)/3 where g(x) is the g.f. of A003080. - Andrew Howroyd, Feb 18 2020

Extended with formula by Christian G. Bower, 10/98

A287889 Number of rooted labeled 4-cactus graphs on 3n+1 nodes.

Original entry on oeis.org

1, 12, 4410, 7560000, 35626991400, 357082280755200, 6536573599765809600, 197543239414923257856000, 9172025443146972656250000000, 619972004905097945232074342400000, 58507834434071888178873434004530400000, 7455351156359319047773396236777475276800000

Offset: 0

N. J. A. Sloane, Jun 21 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..100
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.

Cf. A034940, A287890, A287891, A287892.

[(3*n+1)^n*Factorial(3*n)/(2^n*Factorial(n)): n in [0..12]]; // Vincenzo Librandi, Feb 19 2020

Table[(3 n + 1)^n (3 n)! / (2^n n!), {n, 0, 15}] (* Vincenzo Librandi, Feb 19 2020 *)

seq(n)={my(p=serlaplace(serreverse(x*exp(-x^3/2 + O(x^(3*n+1)))))); vector(n+1, k, polcoef(p, 3*k-2))} \\ Andrew Howroyd, Feb 17 2020

a(n) = (3*n+1)^n*(3*n)!/(2^n*n!). - Andrew Howroyd, Feb 17 2020

a(0) changed and terms a(7) and beyond from Andrew Howroyd, Feb 17 2020

A058927 Numerators of series related to triangular cacti.

Original entry on oeis.org

1, 1, 5, 49, 243, 14641, 371293, 253125, 410338673, 16983563041, 1400846643, 41426511213649, 95367431640625, 617673396283947, 10260628712958602189, 756943935220796320321, 7474615974418932603, 827909024473876953125, 456487940826035155404146917, 510798409623548623605717

Offset: 0

N. J. A. Sloane, Jan 12 2001

From L. Edson Jeffery, Jan 09 2012: (Start)
The reference [Bergeron, et al.] lists the first few terms of the relevant series as S(x) = x + (1/2)*x^3 + (5/8)*x^5 + (49/48)*x^7 + (243/128)*x^9 + ..., from which the numerators were taken for this sequence and the denominators for A058928. This leads to the following
Conjecture: S(x) = Sum_{n>=0} ((2*n+1)^(n-1)/(n!*2^n))*x^(2*n+1) = (A052750(n)/A000165(n))*x^(2*n+1). Letting D_n be the set of divisors of n! and d_n = max(k in D_n : k | (2*n+1)^(n-1)), then a(n)=A052750(n)/d_n. (End)
The above conjecture is correct and follows from formula given in A034940 for the number of rooted labeled triangular cacti with 2n+1 nodes. - Andrew Howroyd, Aug 30 2018

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 307.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000165, A034940, A052750, A058928.

a(n)={numerator((2*n+1)^(n-1)/(2^n*n!))} \\ Andrew Howroyd, Aug 30 2018

G.f.: A(x) satisfies A(x)=exp(x*A(x)^2). - Vladimir Kruchinin, Feb 09 2013

a(n) = numerator(A034940(n)/(2*n+1)!) = numerator((2*n+1)^(n-1)/(2^n*n!)). - Andrew Howroyd, Aug 30 2018

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010

Terms a(12) and beyond from Andrew Howroyd, Aug 30 2018

A058928 Denominators of series related to triangular cacti.

Original entry on oeis.org

1, 2, 8, 48, 128, 3840, 46080, 14336, 10321920, 185794560, 6553600, 81749606400, 78479622144, 209924915200, 1428329123020800, 42849873690624000, 170993385472000, 7611536747003904, 1678343852714360832000, 747740921331712000, 2551082656125828464640000

Offset: 0

N. J. A. Sloane, Jan 12 2001

From L. Edson Jeffery, Jan 09 2012: (Start)
The reference [Bergeron, et al.] lists the first few terms of the relevant series as S(x) = x + (1/2)*x^3 + (5/8)*x^5 + (49/48)*x^7 + (243/128)*x^9 + ..., from which the denominators were taken for this sequence and the numerators for A058927. This leads to the following
Conjecture: S(x) = Sum_{n>=0} ((2*n+1)^(n-1)/(n!*2^n))*x^(2*n+1) = (A052750(n)/A000165(n))*x^(2*n+1). Letting D_n be the set of divisors of n! and d_n = max(k in D_n : k | (2*n+1)^(n-1)), then a(n)=A000165(n)/d_n. (End)
The above conjecture is correct and follows from formula given in A034940 for the number of rooted labeled triangular cacti with 2n+1 nodes. - Andrew Howroyd, Aug 30 2018

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 307.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000165, A034940, A052750, A058927.

a(n)={denominator((2*n+1)^(n-1)/(2^n*n!))} \\ Andrew Howroyd, Aug 30 2018

a(n) = denominator(A034940(n)/(2*n+1)!) = denominator((2*n+1)^(n-1)/(2^n*n!)). - Andrew Howroyd, Aug 30 2018

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010

Terms a(12) and beyond from Andrew Howroyd, Aug 30 2018

A362283 Expansion of e.g.f. exp( sqrt(-LambertW(-x^2)) ).

Original entry on oeis.org

1, 1, 1, 4, 13, 106, 601, 7456, 60649, 1012348, 10748161, 225641296, 2957978101, 74847384184, 1168123938073, 34598428916416, 626497273410961, 21261683280971536, 438222313050326209, 16765636110497697088, 387549609831150094621, 16502188154766906299296

Offset: 0

Seiichi Manyama, Apr 14 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..417
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A000272, A034940, A277614, A362293.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sqrt(-lambertw(-x^2)))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} A034940(k) * binomial(n-1,2*k) * a(n-2*k-1).

A362292 a(n) = (n+1/3)^n * (3*n)!/n!.

Original entry on oeis.org

1, 8, 1960, 2240000, 7037430400, 47023181004800, 573855569801113600, 11561744236447268864000, 357878145625000000000000000, 16126894605060719806880153600000, 1014615361149039162308699493990400000, 86191463407361600308443922239271731200000

Offset: 0

Seiichi Manyama, Apr 14 2023

Winston de Greef, Table of n, a(n) for n = 0..141

Cf. A000169, A034940.

PARI
```
a(n) = (n+1/3)^n*(3*n)!/n!;
```

cp's OEIS Frontend

A034941 Number of labeled triangular cacti with 2n+1 nodes (n triangles).

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A214406 Triangle of second-order Eulerian numbers of type B.

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A003080 Number of rooted triangular cacti with 2n+1 nodes (n triangles).

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A003081 Number of triangular cacti with 2n+1 nodes (n triangles).

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Mathematica

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Extensions

A287889 Number of rooted labeled 4-cactus graphs on 3n+1 nodes.

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Magma

Mathematica

PARI

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A058927 Numerators of series related to triangular cacti.

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A058928 Denominators of series related to triangular cacti.

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