A292186 -id:A292186 - cp's OEIS frontend

A380622 Array read by antidiagonals: T(n,k) is the number of rooted k-regular combinatorial maps with n vertices, n >= 0, k >= 1.

1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 3, 5, 1, 0, 1, 0, 24, 0, 1, 0, 1, 15, 189, 297, 60, 1, 0, 1, 0, 1695, 0, 4896, 0, 1, 0, 1, 105, 19305, 472200, 869400, 100278, 1105, 1, 0, 1, 0, 252000, 0, 242183775, 0, 2450304, 0, 1, 0, 1, 945, 3828825, 2465608950, 103694490900, 198147676875, 16482741030, 69533397, 27120, 1, 0

Offset: 0

Andrew Howroyd, Jan 29 2025

The combinatorial maps considered are connected, unlabeled, may have loops and parallel edges and are of any orientable genus.

			Array begins:
============================================================
n\k | 1 2    3       4      5         6     7          8 ...
----+-------------------------------------------------------
  0 | 1 1    1       1      1         1     1          1 ...
  1 | 0 1    0       3      0        15     0        105 ...
  2 | 1 1    5      24    189      1695 19305     252000 ...
  3 | 0 1    0     297      0    472200     0 2465608950 ...
  4 | 0 1   60    4896 869400 242183775 ...
  5 | 0 1    0  100278      0 ...
  6 | 0 1 1105 2450304 ...
  7 | 0 1    0 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Columns 2..6 are A000012, A062980 (with interspersed zeros), A292186, A380623 (with interspersed zeros), A380624.

Cf. A053979, A269919, A380625.

T(n,k)={my(A=O(x^(n*k+1)), g=serlaplace(serconvol(exp(x^k/k + A), exp(x^2/2 + A)))); polcoef(1 + x*deriv(g)/g, n*k)}

A380625(n) = Sum_{d|2*n} T(d,2*n/d).

A292187 Number of rooted unlabeled bipartite cubic maps on a compact closed oriented surface with 2n vertices (and thus 3n edges), with a(0) = 1.

Original entry on oeis.org

1, 2, 12, 112, 1392, 21472, 394752, 8421632, 204525312, 5572091392, 168331164672, 5585571889152, 201973854584832, 7905697598963712, 333049899230625792, 15025907115679875072, 722841343143300759552, 36935846945562562527232, 1997902532753538016346112, 114050521905958855289864192, 6852141240070150728132329472

Offset: 0

Sasha Kolpakov, Sep 11 2017

Equivalently, the number of rooted bicolored triangulations with 2*n triangles (and thus 3*n edges) for n > 0.
Equivalently, the number of pairs of permutations (alpha,sigma) up to simultaneous conjugacy on a pointed set of size 3*n with alpha^3=sigma^3=1, acting transitively and without fixed points, for n > 0.
This is also the S(3, -5, 1) sequence of Martin and Kearney, if the offset is set to 1.
This sequence is not D-finite (or holonomic).

Sasha Kolpakov, Table of n, a(n) for n = 0..119
Laura Ciobanu and Alexander Kolpakov, Free subgroups of free products and combinatorial hypermaps, Discrete Mathematics, 342 (2019), 1415-1433; arXiv:1708.03842 [math.CO], 2017-2019.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318; arXiv:1103.4936 [math.CO], 2011.

Cf. A000698, A062980, A292186.

from sympy.core.cache import cacheit
@cacheit
def a(n): return 1 if n == 0 else (3*n - 1)*a(n - 1) + sum([a(k)*a(n - k - 1) for k in range(1, n)])
[a(n) for n in range(21)]

a(0)=1, a(1)=2, a(n) = 3*n*a(n-1) + Sum_{k=1..n-2} a(k)*a(n-k-1) for n>=2.
From Peter Bala, Sep 01 2023: (Start)
The o.g.f. A(x) = 1 + 2*x + 12*x^2 + 112*x^3 + 1392*x^4 + 21472*x^5 + 394752*x^6 + ... satisfies the Riccati differential equation (3*x^2)*A'(x) = -1 + (1 - x)*A(x) - x*A(x)^2 with A(0) = 1.
O.g.f. as a continued fraction of Stieltjes type: A(x) = 1/(1 - 2*x/(1 - 4*x/(1 - 5*x/(1 - 7*x/(1 - 8*x/(1 - 10*x/(1 - ... ))))))).
Also A(x) = 1/(1 + 2*x - 4*x/(1 - 2*x/(1 - 7*x/(1 - 5*x/(1 - 10*x/(1 - 8*x/(1 - ... ))))))). (End)

Edited by Andrey Zabolotskiy, Jan 23 2025

A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

Original entry on oeis.org

1, 7, 84, 1463, 33936, 990542, 34938624, 1445713003, 68639375616, 3676366634402, 219208706540544, 14397191399702118, 1032543050697424896, 80280469685284582812, 6725557192852592984064, 603931579625379293509683

Offset: 1

N. J. A. Sloane, Nov 20 2010

nonn

			G.f. = x + 7*x^2 + 84*x^3 + 1463*x^4 + 33936*x^5 + 990542*x^6 + 34938624*x^7 + ...
a(2) = 7 since (6*2 - 4) * a(2-1) - (a(1) * a(2-1)) = 7.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 307.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
NIST Digital Library of Mathematical Functions, Airy Functions.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Eric Weisstein's World of Mathematics, Airy Functions, contains the definitions of Ai(x), Bi(x).

Cf. A000079 S(1,1,-1), A000108 S(0,0,1), A000142 S(1,-1,0), A000244 S(2,1,-2), A000351 S(4,1,-4), A000400 S(5,1,-5), A000420 S(6,1,-6), A000698 S(2,-3,1), A001710 S(1,1,0), A001715 S(1,2,0), A001720 S(1,3,0), A001725 S(1,4,0), A001730 S(1,5,0), A003319 S(1,-2,1), A005411 S(2,-4,1), A005412 S(2,-2,1), A006012 S(-1,2,2), A006318 S(0,1,1), A047891 S(0,2,1), A049388 S(1,6,0), A051604 S(3,1,0), A051605 S(3,2,0), A051606 S(3,3,0), A051607 S(3,4,0), A051608 S(3,5,0), A051609 S(3,6,0), A051617 S(4,1,0), A051618 S(4,2,0), A051619 S(4,3,0), A051620 S(4,4,0), A051621 S(4,5,0), A051622 S(4,6,0), A051687 S(5,1,0), A051688 S(5,2,0), A051689 S(5,3,0), A051690 S(5,4,0), A051691 S(5,5,0), A053100 S(6,1,0), A053101 S(6,2,0), A053102 S(6,3,0), A053103 S(6,4,0), A053104 S(7,1,0), A053105 S(7,2,0), A053106 S(7,3,0), A062980 S(6,-8,1), A082298 S(0,3,1), A082301 S(0,4,1), A082302 S(0,5,1), A082305 S(0,6,1), A082366 S(0,7,1), A082367 S(0,8,1), A105523 S(0,-2,1), A107716 S(3,-4,1), A111529 S(1,-3,2), A111530 S(1,-4,3), A111531 S(1,-5,4), A111532 S(1,-6,5), A111533 S(1,-7,6), A111546 S(1,0,1), A111556 S(1,1,1), A143749 S(0,10,1), A146559 S(1,1,-2), A167872 S(2,-3,2), A172450 S(2,0,-1), A172485 S(-1,-2,3), A177354 S(1,2,1), A292186 S(4,-6,1), A292187 S(3, -5, 1).

a[1] = 1; a[n_]:= a[n] = (6*n-4)*a[n-1] - Sum[a[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Jan 19 2015 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (6 * k - 4) * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */

S(v1, v2, v3, N=16) = {
  my(a = vector(N)); a[1] = 1;
  for (n = 2, N, a[n] = (v1*n+v2)*a[n-1] + v3*sum(j=1,n-1,a[j]*a[n-j])); a;
};
S(6,-4,-1)
\\ test: y = x*Ser(S(6,-4,-1,201)); 6*x^2*y' == y^2 - (2*x-1)*y - x
\\ Gheorghe Coserea, May 12 2017

a(n) = (6*n - 4) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 24 2011
G.f.: x / (1 - 7*x / (1 - 5*x / (1 - 13*x / (1 - 11*x / (1 - 19*x / (1 - 17*x / ... )))))). - Michael Somos, Jan 03 2013
a(n) = 3/(2*Pi^2)*int((4*x)^((3*n-1)/2)/(Ai'(x)^2+Bi'(x)^2), x=0..inf), where Ai'(x), Bi'(x) are the derivatives of the Airy functions. [Vladimir Reshetnikov, Sep 24 2013]
a(n) ~ 6^n * (n-1)! / (2*Pi) [Martin + Kearney, 2011, p.16]. - Vaclav Kotesovec, Jan 19 2015
6*x^2*y' = y^2 - (2*x-1)*y - x, where y(x) = Sum_{n>=1} a(n)*x^n. - Gheorghe Coserea, May 12 2017
G.f.: x/(1 - 2*x - 5*x/(1 - 7*x/(1 - 11*x/(1 - 13*x/(1 - ... - (6*n - 1)*x/(1 - (6*n + 1)*x/(1 - .... Cf. A062980. - Peter Bala, May 21 2017

A292206 Number of unrooted unlabeled connected four-regular maps on a compact closed oriented surface with n vertices (and thus 2*n edges).

Original entry on oeis.org

1, 2, 7, 36, 365, 5250, 103801, 2492164, 70304018, 2265110191, 82013270998, 3295691020635, 145553281837454, 7008046130978980, 365354356543414133, 20504381826687810441, 1232562762503125498772, 79012106044626365750974, 5380476164948914549410335, 387882486153123498708054879

Offset: 0

Sasha Kolpakov, Sep 11 2017

nonn

Equivalently, the number of unrooted quadrangulations of oriented surfaces with n quadrangles (and thus 2*n edges).

Equivalently, the number of pairs (alpha,sigma) of permutations on a set of size 4*n up to simultaneous conjugacy such that alpha (resp. sigma) has only cycles of length 2 (resp. 4) and the subgroup generated by them acts transitively.

			For n = 1, a(n) = 2:
1) the figure-eight map on a sphere (1 vertex, which has degree 4, and 2 edges) <-> its dual map, which is the quadrangulation of a sphere created by a 2-edge path (it bounds 1 region, which has 4 boundary segments, even though they are formed by only 2 different edges) <-> the conjugacy class of the pair of permutations ((12)(34), (1234));
2) the map on a torus consisting of two non-homotopic nontrivial loops (1 vertex, which has degree 4, and 2 edges) <-> its dual map, which is the same map again (it bounds 1 region, which has 4 boundary segments, even though they are formed by only 2 different edges) <-> the conjugacy class of the pair of permutations ((13)(24), (1234)).

Andrew Howroyd, Table of n, a(n) for n = 0..300
Laura Ciobanu and Alexander Kolpakov, Free subgroups of free products and combinatorial hypermaps, Discrete Mathematics, 342 (2019), 1415-1433; arXiv:1708.03842 [math.CO], 2017-2019.

Column 4 of A380626.
Unrooted version of A292186.
Cf. A268556.

Inverse Euler transform of A268556. - Andrew Howroyd, Jan 29 2025

Edited by Andrey Zabolotskiy, Jan 17 2025

a(0)=1 prepended and a(18) onwards from Andrew Howroyd, Jan 29 2025

cp's OEIS Frontend

A380622 Array read by antidiagonals: T(n,k) is the number of rooted k-regular combinatorial maps with n vertices, n >= 0, k >= 1.

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A292187 Number of rooted unlabeled bipartite cubic maps on a compact closed oriented surface with 2*n vertices (and thus 3*n edges), with a(0) = 1.

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A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

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Mathematica

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A292206 Number of unrooted unlabeled connected four-regular maps on a compact closed oriented surface with n vertices (and thus 2*n edges).

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Formula

Extensions

A292187 Number of rooted unlabeled bipartite cubic maps on a compact closed oriented surface with 2n vertices (and thus 3n edges), with a(0) = 1.