Nicolas Bonichon - cp's OEIS frontend

A356197 Number of Baxter 3-permutations of length n.

Original entry on oeis.org

1, 1, 4, 28, 260, 2872, 35620, 479508

Offset: 0

Nicolas Bonichon, Jul 29 2022

Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.

Cf. A001181.

A308526 Number of essentially 3-connected rooted toroidal maps with n vertices.

Original entry on oeis.org

0, 2, 42, 892, 18888, 399280, 8431776, 177936064, 3753206400, 79139040000, 1668268861952, 35160393493504, 740921108899840, 15611120289755136, 328889518650990592, 6928313584957702144, 145939409585973133312, 3073901537848967495680, 64741608434203590524928

Offset: 0

Nicolas Bonichon, Jun 05 2019

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..500
Nicolas Bonichon, Éric Fusy, Benjamin Lévêque, A bijection for essentially 3-connected toroidal maps, arXiv:1907.04016 [math.CO], 2019.

Cf. A308523, A308524, A289208, A006422.

dev_A := 0; n := 20; dev_A := series(RootOf(A-x*(A^2+2*A+2)^2, A), x = 0, n+1): seq(coeff(series(subs(A = dev_A, (1+A)*(A^2+3*A+4)*A/((3*A^2+2*A-2)^2*(A+2))), x, n+1), x, k), k = 0 .. n);

Block[{nn = 19, A, x}, A[] = 0; Do[A[x] = x*(2 + 2*A[x] + A[x]^2)^2 + O[x]^nn, nn]; CoefficientList[(1 + A[x])*(A[x]^2 + 3*A[x] + 4)* A[x]/((3*A[x]^2 + 2*A[x] - 2)^2*(A[x] + 2)), x]] (* Michael De Vlieger, Sep 03 2019 *)

G.f.: (1+A)*(A^2+3*A+4)*A/((3*A^2+2*A-2)^2*(A+2)) where A=x*(2+2*A+A^2)^2.
From Vaclav Kotesovec, Jun 25 2019: (Start)
Recurrence: 81*(n-1)*(3*n - 2)*(3*n - 1)*(20802264*n^8 - 513044308*n^7 + 5457802931*n^6 - 32703730375*n^5 + 120697828661*n^4 - 280851750277*n^3 + 402186188144*n^2 - 323841737040*n + 112137480000)*a(n) = 6*(96792934392*n^11 - 2657166947316*n^10 + 32322659739783*n^9 - 229681592172541*n^8 + 1057798736706708*n^7 - 3309904792738002*n^6 + 7166955104700747*n^5 - 10716261762345309*n^4 + 10816142222455650*n^3 - 6994792735444832*n^2 + 2594496776694720*n - 413761340160000)*a(n-1) - 32*(136213224672*n^11 - 3864805132664*n^10 + 48853431813424*n^9 - 362854015235883*n^8 + 1757540351761182*n^7 - 5820283983972594*n^6 + 13419220917200106*n^5 - 21479458450012897*n^4 + 23298284090559356*n^3 - 16214747993479962*n^2 + 6458737193497260*n - 1099216619550000)*a(n-2) - 768*(29622423936*n^11 - 845570009984*n^10 + 10735773789272*n^9 - 79940670306164*n^8 + 387373872945691*n^7 - 1280558339496068*n^6 + 2940763323423808*n^5 - 4679130395980206*n^4 + 5037190265229413*n^3 - 3476169558457578*n^2 + 1372907413337880*n - 231844115160000)*a(n-3) - 24576*(2*n - 7)*(582463392*n^10 - 14885297224*n^9 + 166341178864*n^8 - 1068833075597*n^7 + 4366030094616*n^6 - 11823901892456*n^5 + 21447449277486*n^4 - 25646549248003*n^3 + 19256170722842*n^2 - 8132937809520*n + 1445811660000)*a(n-4) - 131072*(n-5)*(2*n - 9)*(2*n - 7)*(20802264*n^8 - 346626196*n^7 + 2448956167*n^6 - 9565916473*n^5 + 22545828451*n^4 - 32733304759*n^3 + 28456182418*n^2 - 13430023272*n + 2589840000)*a(n-5).
a(n) ~ (7 + sqrt(7)) * 2^(4*n - 5) * (17 + 7*sqrt(7))^n / 3^(3*n + 1).
(End)

A308524 Number of essentially 3-connected rooted toroidal maps with n edges.

Original entry on oeis.org

0, 0, 1, 2, 11, 40, 166, 658, 2647, 10592, 42446, 169972, 680670, 2725320, 10910992, 43678882, 174843151, 699839680, 2801078662, 11210671612, 44866276906, 179552951440, 718539964132, 2875389341332, 11506176209206, 46042099714240, 184234059839116, 737184620655368

Offset: 0

Nicolas Bonichon, Jun 05 2019

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Nicolas Bonichon, Éric Fusy, Benjamin Lévêque, A bijection for essentially 3-connected toroidal maps, arXiv:1907.04016 [math.CO], 2019.

Cf. A308523, A308526, A289208, A006422.

dev_A := 0; n := 20; dev_A := series(RootOf(A-x*(1+A)^2, A), x = 0, n+1);
seq(coeff(series(subs(A = dev_A, A^2*(1+A)/((1+2*A)*(1-A)^2*(1+3*A))), x, n+1), x, k), k = 0 .. n);
# second Maple program:
a:= proc(n) option remember; `if`(n<6, [0, 0, 1, 2, 11, 40][n+1],
     ((37*n^2-258*n+401)*a(n-1)-6*(2*n^2-25*n+88)*a(n-2)
      -48*(3*n^2-23*n+45)*a(n-3)-32*(n-4)*(2*n-7)*a(n-4))
      /((6*(n-1))*(n-5)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 07 2019

CoefficientList[Series[x*(1 + 8*x + (2*x - 1)*Sqrt[1 - 4*x])/(2*(2 + x)*(1 - 4*x)*(3 + 4*x)), {x, 0, 30}], x] (* Vaclav Kotesovec, Jun 25 2019 *)

G.f.: A^2*(1+A)/((1+2*A)*(1-A)^2*(1+3*A)) where A=x*(1+A)^2.
G.f.: x*(1 + 8*x + (2*x - 1)*sqrt(1 - 4*x))/(2*(2 + x)*(1 - 4*x)*(3 + 4*x)). - Vaclav Kotesovec, Jun 25 2019
a(n) ~ 2^(2*n - 3) / 3. - Vaclav Kotesovec, Jun 25 2019

A308523 Number of essentially simple rooted toroidal triangulations with n vertices.

Original entry on oeis.org

0, 1, 10, 97, 932, 8916, 85090, 810846, 7719048, 73431340, 698187400, 6635738209, 63047912372, 598885073788, 5687581936284, 54005562798252, 512728901004816, 4867263839614716, 46199494669833400, 438481077306427924, 4161316466910824272

Offset: 0

Nicolas Bonichon, Jun 05 2019

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..499
Nicolas Bonichon, Éric Fusy, Benjamin Lévêque, A bijection for essentially 3-connected toroidal maps, arXiv:1907.04016 [math.CO], 2019.
Éric Fusy, Benjamin Lévêque, Orientations and bijections for toroidal maps with prescribed face-degrees and essential girth, arXiv:1807.00522 [math.CO], 2018. See Proposition 25 p. 37.

Cf. A002293, A308524, A308526, A289208, A006422.
Cf. A000302, A036829.
Cf. A386565, A386611, A386612.

n:=20:
dev_A := series(RootOf(A-x*(1+A)^4, A), x = 0, n+1):
seq(coeff(series(subs(A=dev_A, A/(1-3*A)^2), x, n+1), x, k), k=0..n);

terms = 21;
A[] = 0; Do[A[x] = x (1 + A[x])^4 + O[x]^terms, terms];
CoefficientList[A[x]/(1 - 3 A[x])^2, x] (* Jean-François Alcover, Jun 17 2019 *)

my(N=30, x='x+O('x^N), g=x*sum(k=0, N, binomial(4*k+2, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-4*g)^2)) \\ Seiichi Manyama, Jul 19 2025

G.f.: A/(1-3*A)^2 where A=x(1+A)^4 is the g.f. of A002293.
From Vaclav Kotesovec, Jun 25 2019: (Start)
Recurrence: 81*(n-1)*(3*n - 2)*(3*n - 1)*(24*n - 37)*a(n) = 24*(13824*n^4 - 59328*n^3 + 92832*n^2 - 62278*n + 14653)*a(n-1) - 2048*(2*n - 3)*(4*n - 7)*(4*n - 5)*(24*n - 13)*a(n-2).
a(n) ~ 2^(8*n - 3) / 3^(3*n). (End)
From Seiichi Manyama, Jul 19 2025: (Start)
G.f.: g*(1-g)/(1-4*g)^2 where g*(1-g)^3 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/4) * log( Sum_{k>=0} binomial(4*k,k)*x^k ). (End)
From Seiichi Manyama, Jul 28 2025: (Start)
a(n) = Sum_{k=0..n-1} binomial(4*k-2+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n-1,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k-1,k). (End)

A289208 Number of rooted essentially 4-connected toroidal triangulations with n vertices.

Original entry on oeis.org

0, 1, 6, 40, 268, 1801, 12120, 81628, 550040, 3707635, 24997966, 168573824, 1136933488, 7668785996, 51731557296, 348991600660, 2354505179952, 15885669341751, 107183855819490, 723217053276952, 4880016412621148, 32929530655094281

Offset: 0

Nicolas Bonichon, Jun 28 2017

nonn

N. Bonichon and B. Lévêque, A bijection for essentially 4-connected toroidal triangulations, arXiv preprint arXiv:1707.08191 [cs.DM], 2017.
N. Bonichon and B. Lévêque, A bijection for essentially 4-connected toroidal triangulations, The Electronic Journal of Combinatorics 26 (2019) P1.13.

Cf. A001764.

n := 30; t := series(RootOf(729*T^3*x^3+2700*T^3*x^2-848*T^3*x +756*T^2*x^2 +64*T^3 -112*T^2*x +54*T*x^2-T*x+x^2, T), x = 0, n+1): seq(coeff(t, x, k), k = 0 .. n);

terms = 22; T[] = 0; Do[T[x] = (1/(x (-1 + 54 x)))(-x^2 + 112 x T[x]^2 - 756 x^2 T[x]^2 - 64 T[x]^3 + 848 x T[x]^3 - 2700 x^2 T[x]^3 - 729 x^3 T[x]^3) + O[x]^terms // Normal, {terms}];
CoefficientList[T[x], x] (* Jean-François Alcover, Nov 16 2018 *)

G.f.: x*A/(7*A^2*x - 21*A*x + 9*x + 1) where A = 1+x*A^3 is the g.f. of A001764.
0 = 729*T^3*x^3 + 2700*T^3*x^2 - 848*T^3*x + 756*T^2*x^2 + 64*T^3 - 112*T^2*x + 54*T*x^2 - T*x + x^2 where T is the g.f. of this sequence.
From Vaclav Kotesovec, Jun 25 2019: (Start)
a(n) ~ 3^(3*n) / 2^(2*n + 3).
Recurrence: 32*(n-1)*(2*n-1)*(3*n-1)*(7*n-18)*a(n) = 16*(1113*n^4 - 5753*n^3 + 8619*n^2 - 1717*n - 3462)*a(n-1) - 6*(9450*n^4 - 56367*n^3 + 93156*n^2 - 2813*n - 64226)*a(n-2) - 81*(3*n-8)*(3*n-7)*(3*n+2)*(7*n-11)*a(n-3). (End)

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User: Nicolas Bonichon

A356197 Number of Baxter 3-permutations of length n.

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A308526 Number of essentially 3-connected rooted toroidal maps with n vertices.

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A308524 Number of essentially 3-connected rooted toroidal maps with n edges.

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A308523 Number of essentially simple rooted toroidal triangulations with n vertices.

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A289208 Number of rooted essentially 4-connected toroidal triangulations with n vertices.

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