A006422 -id:A006422 - cp's OEIS frontend

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

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)

A343090 Triangle read by rows: T(n,k) is the number of rooted toroidal maps with n edges and k faces and without separating cycles or isthmuses, n >= 2, k = 1..n-1.

Original entry on oeis.org

1, 4, 4, 10, 47, 10, 20, 240, 240, 20, 35, 831, 2246, 831, 35, 56, 2282, 12656, 12656, 2282, 56, 84, 5362, 52164, 109075, 52164, 5362, 84, 120, 11256, 173776, 648792, 648792, 173776, 11256, 120, 165, 21690, 495820, 2978245, 5360286, 2978245, 495820, 21690, 165

Offset: 2

Andrew Howroyd, Apr 04 2021

The number of vertices is n-k.

Column k is a polynomial of degree 3*k. This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.

			Triangle begins:
    1;
    4,     4;
   10,    47,     10;
   20,   240,    240,     20;
   35,   831,   2246,    831,     35;
   56,  2282,  12656,  12656,   2282,     56;
   84,  5362,  52164, 109075,  52164,   5362,    84;
  120, 11256, 173776, 648792, 648792, 173776, 11256, 120;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..1276
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, Table VIc.

Columns 1..4 are A000292, A006422, A006423, A006424.
Row sums are A343091.
Cf. A269921, A342980, A342989, A343092.

\\ Needs F from A342989.
G(n,m,y,z)={my(p=F(n,m,y,z)); subst(p, x, serreverse(x*p^2))}
H(n, g=1)={my(q=G(n, g, 'y, 'z)-x*(1+'z), v=Vec(polcoef(sqrt(serreverse(x/q^2)/x), g, 'y))); [Vecrev(t) | t<-v]}
{ my(T=H(10)); for(n=1, #T, print(T[n])) }

T(n,n-k) = T(n,k).

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

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)

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

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A343090 Triangle read by rows: T(n,k) is the number of rooted toroidal maps with n edges and k faces and without separating cycles or isthmuses, n >= 2, k = 1..n-1.

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

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

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