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

Author

Andrew Howroyd, Jan 29 2025

Keywords

• nonn
• tabl

Comments

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

Examples

			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 ...
  ...
		

Links

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

Crossrefs

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

Cf. A053979, A269919, A380625.

Programs

PARI

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)}

Formula

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