A034941 -id:A034941 - cp's OEIS frontend

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

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

A287890 Number of unrooted labeled 4-cactus graphs on 3n+1 nodes.

Original entry on oeis.org

1, 3, 630, 756000, 2740537800, 22317642547200, 344030189461358400, 8979238155223784448000, 366881017725878906250000000, 22141857318039212329716940800000, 1887349497873286715447530129178400000, 219275034010568207287452830493455155200000

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. A034941, A287889, A287891, A287892.

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

Table[(3 n + 1)^(n-1) (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))))/x)); vector(n+1, k, polcoef(p, 3*k-3))} \\ Andrew Howroyd, Feb 17 2020

a(n) = (3*n+1)^(n-1)*(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

A361355 Triangle read by rows: T(n,k) is the number of simple series-parallel matroids on [n] with rank k, 1 <= k <= n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 15, 1, 0, 0, 0, 0, 75, 1, 0, 0, 0, 0, 735, 280, 1, 0, 0, 0, 0, 0, 9345, 938, 1, 0, 0, 0, 0, 0, 76545, 77805, 2989, 1, 0, 0, 0, 0, 0, 0, 1865745, 536725, 9285, 1, 0, 0, 0, 0, 0, 0, 13835745, 27754650, 3334870, 28446, 1, 0

Offset: 1

Andrew Howroyd, Mar 09 2023

			Triangle begins:
  1;
  0, 0;
  0, 1,  0;
  0, 0,  1,   0;
  0, 0, 15,   1,     0;
  0, 0,  0,  75,     1,     0;
  0, 0,  0, 735,   280,     1,    0;
  0, 0,  0,   0,  9345,   938,    1, 0;
  0, 0,  0,   0, 76545, 77805, 2989, 1, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Luis Ferroni and Matt Larson, Kazhdan-Lusztig polynomials of braid matroids, arXiv:2303.02253 [math.CO], 2023.
Nicholas Proudfoot, Yuan Xu, and Ben Young, On the enumeration of series-parallel matroids, arXiv:2406.04502 [math.CO], 2024.

Row sums are A007834.
Cf. A140945, A359985, A361353.
Cf. A034941, A361282.

\\ B gives A359985 as e.g.f.
B(n)= {exp(x*(1+y) + y*intformal(serreverse(log(1 + x*y + O(x^n))/y + log(1 + x + O(x^n)) - x)))}
T(n) = {my(v=Vec(serlaplace(log(subst(B(n), x, log(1 + x + O(x*x^n)))/(1 + x))))); vector(#v, n, Vecrev(v[n]/y, n))}
{ my(A=T(9)); for(i=1, #A, print(A[i])) }

E.g.f.: A(x,y) = log(1 + B(x,y)) where B(x,y) is the e.g.f. of A361353.
E.g.f.: A(x,y) = log(B(log(1 + x), y)/(1 + x)) where B(x,y) is the e.g.f. of A359985.
T(2*n+1, n+1) = A034941(n).
T(2*n, n+1) = A361282(n).

A200313 E.g.f. satisfies: A(x) = exp(x^3*A(x)^3/3!).

Original entry on oeis.org

1, 1, 70, 28000, 33833800, 91842150400, 471920698849600, 4105733038511104000, 55918460253906250000000, 1124922893768186370457600000, 31962429471680921191680301600000, 1237813985055170041194334820761600000, 63474917512551971525535771981021376000000

Offset: 0

Paul D. Hanna, Nov 15 2011

nonn

			E.g.f.: A(x) = 1 + x^3/3! + 70*x^6/6! + 28000*x^9/9! + 33833800*x^12/12! + ...
where log(A(x)) = x^3*A(x)^3/3! and
A(x)^3 = 1 + 3*x^3/3! + 270*x^6/6! + 120960*x^9/9! + 155925000*x^12/12! + ...

G. C. Greubel, Table of n, a(n) for n = 0..100

Cf. A025035, A034941, A200314, A200315.

List([0..10],n->(3*n+1)^(n-1)*Factorial(3*n)/(Factorial(n)*Factorial(3)^n)); # Muniru A Asiru, Jul 28 2018

[(3*n+1)^(n-1)*Factorial(3*n)/(6^n*Factorial(n)): n in [0..30]]; // G. C. Greubel, Jul 27 2018

Table[(3*n + 1)^(n - 1)*(3*n)!/(n!*(3!)^n), {n, 0, 30}] (* G. C. Greubel, Jul 27 2018 *)

{a(n)=(3*n)!*polcoeff(1/x*serreverse(x*(exp(-x^3/3!+x*O(x^(3*n))))),3*n)}

{a(n)=(3*n+1)^(n-1)*(3*n)!/(n!*(3!)^n)}

a(n) = (3*n+1)^(n-1) * (3*n)!/(n!*(3!)^n).
E.g.f.: (1/x)*Series_Reversion( x*exp(-x^3/3!) ).
Powers of e.g.f.: define a(n,m) by A(x)^m = Sum_{n>=0} a(n,m)*x^(3*n)/(3*n)!
then a(n,m) = m*(3*n+m)^(n-1) * (3*n)!/(n!*(3!)^n).

A200314 E.g.f. satisfies: A(x) = exp(x^4*A(x)^4/4!).

Original entry on oeis.org

1, 1, 315, 975975, 12909521625, 495181420358625, 44035787449951171875, 7845481113748784765634375, 2526730187976408357560632640625, 1362965093449949100037985665872890625, 1160978904909328561005478318639484556796875

Offset: 0

Paul D. Hanna, Nov 15 2011

nonn

			E.g.f.: A(x) = 1 + x^4/4! + 315*x^8/8! + 975975*x^12/12! + ...
where log(A(x)) = x^4*A(x)^4/4! and
A(x)^4 = 1 + 4*x^4/4! + 1680*x^8/8! + 5913600*x^12/12! + 84084000000*x^16/16! + ...

G. C. Greubel, Table of n, a(n) for n = 0..100

Cf. A034941, A200313, A200315.

List([0..10],n->(4*n+1)^(n-1)*Factorial(4*n)/(Factorial(n)*Factorial(4)^n)); # Muniru A Asiru, Jul 28 2018

[(4*n+1)^(n-1)*Factorial(4*n)/(24^n*Factorial(n)): n in [0..30]]; // G. C. Greubel, Jul 27 2018

Table[(4*n + 1)^(n - 1)*(4*n)!/(n!*(4!)^n), {n, 0, 30}] (* G. C. Greubel, Jul 27 2018 *)

{a(n)=(4*n)!*polcoeff(1/x*serreverse(x*(exp(-x^4/4!+x*O(x^(4*n))))),4*n)}

{a(n)=(4*n+1)^(n-1)*(4*n)!/(n!*(4!)^n)};

a(n) = (4*n+1)^(n-1) * (4*n)!/(n!*(4!)^n).
E.g.f.: (1/x)*Series_Reversion( x*exp(-x^4/4!) ).
Powers of e.g.f.: define a(n,m) by A(x)^m = Sum_{n>=0} a(n,m)*x^(4*n)/(4*n)!
then a(n,m) = m*(4*n+m)^(n-1) * (4*n)!/(n!*(4!)^n).

A200315 E.g.f. satisfies: A(x) = exp(x^5*A(x)^5/5!).

Original entry on oeis.org

1, 1, 1386, 32288256, 4527372986136, 2373840824586206976, 3532226719132271834449776, 12455133709483299692008910094336, 91656142095228409912231665590704016256, 1280796898530759870923631204720457656538791936

Offset: 0

Paul D. Hanna, Nov 15 2011

nonn

			E.g.f.: A(x) = 1 + x^5/5! + 1386*x^10/10! + 32288256*x^15/15! +...
where log(A(x)) = x^5*A(x)^5/5! and
A(x)^5 = 1 + 5*x^5/5! + 9450*x^10/10! + 252252000*x^15/15! + 38192529375000*x^20/20! +...

Cf. A034941, A200313, A200314.

{a(n)=(5*n)!*polcoeff(1/x*serreverse(x*(exp(-x^5/5!+x*O(x^(5*n))))),5*n)}

PARI
```
{a(n)=(5*n+1)^(n-1)*(5*n)!/(n!*(5!)^n)}
```

a(n) = (5*n+1)^(n-1) * (5*n)! / (n!*(5!)^n).
E.g.f.: (1/x)*Series_Reversion( x*exp(-x^5/5!) ).
Powers of e.g.f.: define a(n,m) by A(x)^m = Sum_{n>=0} a(n,m)*x^(5*n)/(5*n)!
then a(n,m) = m*(5*n+m)^(n-1) * (5*n)!/(n!*(5!)^n).

A361282 Number of rank n+1 simple connected series-parallel matroids on [2n].

Original entry on oeis.org

0, 1, 75, 9345, 1865745, 554479695, 231052877055, 128938132548225, 92986310399407425, 84250567868935042575, 93744545254140599193375, 125717783386887888296925825, 200041202339679732328342670625, 372688996228146502285257581079375, 803768398459351988653830600415029375

Offset: 1

Matt Larson, Mar 06 2023

nonn

			For n=2 the a(2) = 1 rank 3 simple connected series-parallel matroid on [4] is the uniform matroid of rank 3.

Luis Ferroni and Matt Larson, Kazhdan-Lusztig polynomials of braid matroids, arXiv:2303.02253 [math.CO], 2023.

Cf. A034941, A361355.

a(n) = T(2*n)[2*n][n+1] \\ T(n) defined in A361355. - Andrew Howroyd, Mar 09 2023

Terms a(10)-a(15) from Andrew Howroyd, Mar 09 2023

A338280 Triangle T read by rows: T(n, k) = k*n^(n-k-1) with 0 < k < n.

Original entry on oeis.org

1, 3, 2, 16, 8, 3, 125, 50, 15, 4, 1296, 432, 108, 24, 5, 16807, 4802, 1029, 196, 35, 6, 262144, 65536, 12288, 2048, 320, 48, 7, 4782969, 1062882, 177147, 26244, 3645, 486, 63, 8, 100000000, 20000000, 3000000, 400000, 50000, 6000, 700, 80, 9, 2357947691, 428717762, 58461513, 7086244, 805255, 87846, 9317, 968, 99, 10

Offset: 2

Stefano Spezia, Oct 20 2020

T(n, k) is the number of forests of n - k edges that connect every other labeled vertex to one of the k roots (see Section 3 in Wästlund).

Alfred Rényi, Some remarks on the theory of trees. MTA Mat. Kut. Inst. Kozl. (Publ. math. Inst. Hungar. Acad. Sci) 4 (1959), 73-85.

Arthur Cayley, A theorem on trees, Quart. J. Pure Appl. Math. 23: 376-378 (1889). Also in The collected mathematical papers of Arthur Cayley vol 13.
John Riordan, Forests of labeled trees, Journal of Combinatorial Theory 5 (1968), 93-103.
Lajos Takács, On Cayley’s Formula for Counting Forests, Journal of Combinatorial Theory Series A 53, 321-323 (1990). See Equation 1.
Johan Wästlund, Padlock Solitaire: A martingale trick for combinatorial enumeration, arXiv:2008.13017 [math.CO], 2020. See Section 3.

Cf. A000027 (diagonal), A000169, A000272 (1st column), A000312, A007334 (2nd column), A023811 (row sums), A034941, A072590, A075363, A210725.

Table[k*n^(n-k-1),{n,2,11},{k,1,n-1}]//Flatten

cp's OEIS Frontend

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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Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A287890 Number of unrooted labeled 4-cactus graphs on 3n+1 nodes.

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Magma

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PARI

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A361355 Triangle read by rows: T(n,k) is the number of simple series-parallel matroids on [n] with rank k, 1 <= k <= n.

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A200313 E.g.f. satisfies: A(x) = exp(x^3*A(x)^3/3!).

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A200314 E.g.f. satisfies: A(x) = exp(x^4*A(x)^4/4!).

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GAP

Magma

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PARI

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A200315 E.g.f. satisfies: A(x) = exp(x^5*A(x)^5/5!).

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