A386565 -id:A386565 - 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)

A386612 a(n) = Sum_{k=0..n-1} binomial(4k+1,k) binomial(4n-4k,n-k-1).

Original entry on oeis.org

0, 1, 13, 142, 1464, 14689, 145154, 1420812, 13818784, 133793940, 1291073809, 12426782294, 119371355672, 1144851458526, 10965655515588, 104919037771224, 1002960800712720, 9580390527192940, 91453374122574372, 872513477065735768, 8320168165323802464, 79305962393873976417

Offset: 0

Seiichi Manyama, Jul 27 2025

nonn

Cf. A078995, A308523, A386565, A386611.

Cf. A006419, A386614, A386616, A386617.

a(n) = sum(k=0, n-1, binomial(4*k+1, k)*binomial(4*n-4*k, n-k-1));

G.f.: g^3 * (g-1)/(4-3*g)^2 where g=1+x*g^4.
G.f.: g/((1-g)^2 * (1-4*g)^2) where g*(1-g)^3 = x.
a(n) = Sum_{k=0..n-1} binomial(4*k+1+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+2,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k+2,k).
D-finite with recurrence 13122*n*(3*n+2)*(3*n+1)*a(n) +81*(-124803*n^3+284553*n^2-210740*n+42140)*a(n-1) +24*(7476768*n^3-29253744*n^2+37920106*n-16562575)*a(n-2) +40960*(-26344*n^3+148032*n^2-282329*n+185874)*a(n-3) +55050240*(2*n-5)*(4*n-13)*(4*n-11)*a(n-4)=0. - R. J. Mathar, Aug 10 2025

A386566 a(n) = Sum_{k=0..n-1} binomial(5k-1,k) binomial(5n-5k,n-k-1).

Original entry on oeis.org

0, 1, 14, 181, 2284, 28506, 353630, 4370584, 53882392, 663116347, 8150224204, 100073884670, 1227826127020, 15055154471696, 184508186225552, 2260299193652496, 27679951219660080, 338872887728053465, 4147618793911034330, 50753529798492061819, 620942367878256638264

Offset: 0

Seiichi Manyama, Jul 26 2025

nonn

			(1/4) * log( Sum_{k>=0} binomial(5*k-1,k)*x^k ) = x + 7*x^2 + 181*x^3/3 + 571*x^4 + 28506*x^5/5 + ...

Cf. A000346, A062236, A386565, A386567.
Cf. A079678, A386367, A386613, A386614.
Cf. A002294, A118971.

a(n) = sum(k=0, n-1, binomial(5*k-1, k)*binomial(5*n-5*k, n-k-1));

my(N=30, x='x+O('x^N), g=sum(k=0, N, binomial(5*k, k)/(4*k+1)*x^k)); concat(0, Vec(g*(g-1)/(5-4*g)^2))

G.f.: g*(g-1)/(5-4*g)^2 where g=1+x*g^5.
G.f.: g/(1-5*g)^2 where g*(1-g)^4 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/4) * log( Sum_{k>=0} binomial(5*k-1,k)*x^k ).
a(n) = Sum_{k=0..n-1} binomial(5*k-1+l,k) * binomial(5*n-5*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n,k).
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k,k).
Conjecture D-finite with recurrence 196608*n*(4*n-3)*(2*n-1)*(18270873280*n -32560150837) *(4*n-1)*a(n) +1280*(-1399185802400000*n^5 +1022280893000000*n^4 +17669158913120000*n^3 -48968110172924750*n^2 +49502057719349955*n -17877514345852392)*a(n-1) +125000*(-61298198200000*n^5 +1447969779032500*n^4 -7721498995066250*n^3 +17474948768595875*n^2 -18352567310653770*n +7399184154389181)*a(n-2) +48828125*(5*n-11) *(5*n-14)*(4958243695*n -6717884799) *(5*n-13)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

A386567 a(n) = Sum_{k=0..n-1} binomial(6k-1,k) binomial(6n-6k,n-k-1).

Original entry on oeis.org

0, 1, 17, 268, 4129, 62955, 954392, 14417376, 217279857, 3269099590, 49125066135, 737516631908, 11064270530632, 165889863957065, 2486052264852180, 37241727274394640, 557707191712371729, 8349517132932620730, 124971965902300790390, 1870139909398530770760

Offset: 0

Seiichi Manyama, Jul 26 2025

nonn

			(1/5) * log( Sum_{k>=0} binomial(6*k-1,k)*x^k ) = x + 17*x^2/2 + 268*x^3/3 + 4129*x^4/4 + 12591*x^5 + ...

Cf. A000346, A062236, A386565, A386566.
Cf. A079679, A386368, A386615, A386616.
Cf. A002295, A130564.

a(n) = sum(k=0, n-1, binomial(6*k-1, k)*binomial(6*n-6*k, n-k-1));

my(N=20, x='x+O('x^N), g=sum(k=0, N, binomial(6*k, k)/(5*k+1)*x^k)); concat(0, Vec(g*(g-1)/(6-5*g)^2))

G.f.: g*(g-1)/(6-5*g)^2 where g=1+x*g^6.
G.f.: g/(1-6*g)^2 where g*(1-g)^5 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/5) * log( Sum_{k>=0} binomial(6*k-1,k)*x^k ).
a(n) = Sum_{k=0..n-1} binomial(6*k-1+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n,k).
a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k,k).

A386611 a(n) = Sum_{k=0..n-1} binomial(4k,k) binomial(4n-4k,n-k-1).

Original entry on oeis.org

0, 1, 12, 126, 1268, 12513, 122148, 1184364, 11432100, 109997460, 1055891248, 10117633542, 96812495820, 925334377822, 8836315646616, 84317468847768, 804064275489924, 7663595943744876, 73009005101019792, 695263276434909976, 6618709687608909648, 62989317586872238689

Offset: 0

Seiichi Manyama, Jul 27 2025

nonn

Cf. A308523, A386565, A386612.

Cf. A008549, A386613, A386615.

a(n) = sum(k=0, n-1, binomial(4*k, k)*binomial(4*n-4*k, n-k-1));

G.f.: g^2 * (g-1)/(4-3*g)^2 where g=1+x*g^4.
G.f.: g/((1-g) * (1-4*g)^2) where g*(1-g)^3 = x.
a(n) = Sum_{k=0..n-1} binomial(4*k+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).

A385251 a(n) = Sum_{k=0..n-1} binomial(4k-3,k) binomial(4n-4k,n-k-1).

Original entry on oeis.org

0, 1, 9, 84, 790, 7452, 70401, 665692, 6298236, 59612556, 564393460, 5344664400, 50621130078, 479513718116, 4542730477758, 43039907282664, 407809863233592, 3864303038901996, 36619104142640460, 347027703183853552, 3288802989845088504, 31169274939274755312

Offset: 0

Seiichi Manyama, Jul 28 2025

nonn

Cf. A078995, A308523, A386565, A386611, A386612.

a(n) = sum(k=0, n-1, binomial(4*k-3, k)*binomial(4*n-4*k, n-k-1));

G.f.: (g-1)/(g * (4-3*g)^2) where g=1+x*g^4.
G.f.: g * (1-g)^2/(1-4*g)^2 where g*(1-g)^3 = x.
a(n) = Sum_{k=0..n-1} binomial(4*k-3+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-2,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k-2,k).

cp's OEIS Frontend

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

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A386612 a(n) = Sum_{k=0..n-1} binomial(4*k+1,k) * binomial(4*n-4*k,n-k-1).

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A386566 a(n) = Sum_{k=0..n-1} binomial(5*k-1,k) * binomial(5*n-5*k,n-k-1).

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A386567 a(n) = Sum_{k=0..n-1} binomial(6*k-1,k) * binomial(6*n-6*k,n-k-1).

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A386611 a(n) = Sum_{k=0..n-1} binomial(4*k,k) * binomial(4*n-4*k,n-k-1).

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A385251 a(n) = Sum_{k=0..n-1} binomial(4*k-3,k) * binomial(4*n-4*k,n-k-1).

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A386612 a(n) = Sum_{k=0..n-1} binomial(4k+1,k) binomial(4n-4k,n-k-1).

A386566 a(n) = Sum_{k=0..n-1} binomial(5k-1,k) binomial(5n-5k,n-k-1).

A386567 a(n) = Sum_{k=0..n-1} binomial(6k-1,k) binomial(6n-6k,n-k-1).

A386611 a(n) = Sum_{k=0..n-1} binomial(4k,k) binomial(4n-4k,n-k-1).

A385251 a(n) = Sum_{k=0..n-1} binomial(4k-3,k) binomial(4n-4k,n-k-1).