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%I #16 Dec 21 2023 02:58:58
%S 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,
%T 10,1,0,110410,20628,3426,568,90,12,1,0,1708394,273064,38886,5696,850,
%U 120,14,1
%N Triangle read by rows. Number of perfect matchings by number of connected components.
%C The exact definition is given in Sokal and Zeng. See section 4.4 and theorem 4.6.
%H Alan D. Sokal and Jiang Zeng, <a href="https://doi.org/10.1016/j.aam.2022.102341">Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions</a>, Advances in Applied Mathematics, Volume 138, 2022. Table on p. 91.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Perfect_matching">Perfect matching</a>.
%F 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
%e Table T(n, k) begins:
%e [0] 1;
%e [1] 0, 1;
%e [2] 0, 2, 1;
%e [3] 0, 10, 4, 1;
%e [4] 0, 74, 24, 6, 1;
%e [5] 0, 706, 188, 42, 8, 1;
%e [6] 0, 8162, 1808, 350, 64, 10, 1;
%e [7] 0, 110410, 20628, 3426, 568, 90, 12, 1;
%e [8] 0, 1708394, 273064, 38886, 5696, 850, 120, 14, 1;
%Y Cf. A001147 (row sums), A000698 (indecomposable perfect matchings), A177797.
%Y T(n,0) = A000007(n), T(n,1) = A000698(n) assuming offset 1.
%K nonn,tabl,more
%O 0,5
%A _Peter Luschny_, Apr 15 2023