A362308 Triangle read by rows. Number of perfect matchings by number of connected components.

1, 0, 1, 0, 2, 1, 0, 10, 4, 1, 0, 74, 24, 6, 1, 0, 706, 188, 42, 8, 1, 0, 8162, 1808, 350, 64, 10, 1, 0, 110410, 20628, 3426, 568, 90, 12, 1, 0, 1708394, 273064, 38886, 5696, 850, 120, 14, 1

Offset

0,5

Comments

The exact definition is given in Sokal and Zeng. See section 4.4 and theorem 4.6.

Links

Table of n, a(n) for n=0..44.

Alan D. Sokal and Jiang Zeng, Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions, Advances in Applied Mathematics, Volume 138, 2022. Table on p. 91.

Wikipedia, Perfect matching.

Formula

T(n, k) = T(n, k-1) - T(n-1, k-2) - (2*n - k - 1)/(k - 1) * T(n - 1, k - 1) for k > 1. - Detlef Meya, Dec 21 2023

Example

Table T(n, k) begins:
  [0] 1;
  [1] 0,       1;
  [2] 0,       2,      1;
  [3] 0,      10,      4,     1;
  [4] 0,      74,     24,     6,    1;
  [5] 0,     706,    188,    42,    8,   1;
  [6] 0,    8162,   1808,   350,   64,  10,   1;
  [7] 0,  110410,  20628,  3426,  568,  90,  12,  1;
  [8] 0, 1708394, 273064, 38886, 5696, 850, 120, 14, 1;

Crossrefs

Cf. A001147 (row sums), A000698 (indecomposable perfect matchings), A177797.

T(n,0) = A000007(n), T(n,1) = A000698(n) assuming offset 1.

Sequence in context: A115563 A364068 A293881 * A185285 A268434 A010107

Adjacent sequences: A362305 A362306 A362307 * A362309 A362310 A362311

Keyword

nonn,tabl,more

Author

Peter Luschny, Apr 15 2023

Status

approved