A351825 -id:A351825 - cp's OEIS frontend

A351823 Triangular array read by rows. T(n,k) is the number of sets of lists (as in A000262(n)) with exactly k size 2 lists, n >= 0, 0 <= k <= floor(n/2).

1, 1, 1, 2, 7, 6, 49, 12, 12, 301, 140, 60, 2281, 1470, 180, 120, 21211, 12642, 2940, 840, 220417, 127736, 41160, 3360, 1680, 2528569, 1527192, 455112, 70560, 15120, 32014801, 19837530, 5748120, 1234800, 75600, 30240, 442974511, 278142590, 83995560, 16687440, 1940400, 332640

Offset: 0

Geoffrey Critzer, Feb 20 2022

From the asymptotic estimate of A000262(n) provided by Vaclav Kotesovec we deduce that in the limit as n gets big the average number of size 2 lists is equal to 1. In other words, lim_{n->oo} Sum_{k>=1} T(n,k)*k/A000262(n) = 1. Generally for any j >= 1, the average number of size j lists equals 1 in the limit as n -> oo.

			Triangle T(n,k) begins:
      1;
      1;
      1,     2;
      7,     6;
     49,    12,   12;
    301,   140,   60;
   2281,  1470,  180, 120;
  21211, 12642, 2940, 840;
  ...

Column k=1 gives A113235.
T(n,floor(n/2)) gives A081125.
T(2n,n) gives A001813.
Cf. A000262 (row sums) A006152, A114329, A351825.

b:= proc(n) option remember; expand(`if`(n=0, 1, add(j!*
     `if`(j=2, x, 1)*b(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..12);  # Alois P. Heinz, Feb 20 2022

nn = 7; Map[Select[#, # > 0 &] &,Range[0, nn]! CoefficientList[Series[Exp[ x/(1 - x) - x ^2 + y x^2], {x, 0, nn}], {x, y}]] // Grid

E.g.f.: exp(x/(1-x) - x^2 + y*x^2).

Sum_{k=0..floor(n/2)} k * T(n,k) = A351825(n). - Alois P. Heinz, Feb 24 2022

cp's OEIS Frontend

A351823 Triangular array read by rows. T(n,k) is the number of sets of lists (as in A000262(n)) with exactly k size 2 lists, n >= 0, 0 <= k <= floor(n/2).

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