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).

%I A351823 #26 Jan 20 2023 01:31:32
%S A351823 1,1,1,2,7,6,49,12,12,301,140,60,2281,1470,180,120,21211,12642,2940,
%T A351823 840,220417,127736,41160,3360,1680,2528569,1527192,455112,70560,15120,
%U A351823 32014801,19837530,5748120,1234800,75600,30240,442974511,278142590,83995560,16687440,1940400,332640
%N 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).
%C A351823 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.
%F A351823 E.g.f.: exp(x/(1-x) - x^2 + y*x^2).
%F A351823 Sum_{k=0..floor(n/2)} k * T(n,k) = A351825(n). - _Alois P. Heinz_, Feb 24 2022
%e A351823 Triangle T(n,k) begins:
%e A351823       1;
%e A351823       1;
%e A351823       1,     2;
%e A351823       7,     6;
%e A351823      49,    12,   12;
%e A351823     301,   140,   60;
%e A351823    2281,  1470,  180, 120;
%e A351823   21211, 12642, 2940, 840;
%e A351823   ...
%p A351823 b:= proc(n) option remember; expand(`if`(n=0, 1, add(j!*
%p A351823      `if`(j=2, x, 1)*b(n-j)*binomial(n-1, j-1), j=1..n)))
%p A351823     end:
%p A351823 T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
%p A351823 seq(T(n), n=0..12);  # _Alois P. Heinz_, Feb 20 2022
%t A351823 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
%Y A351823 Column k=1 gives A113235.
%Y A351823 T(n,floor(n/2)) gives A081125.
%Y A351823 T(2n,n) gives A001813.
%Y A351823 Cf. A000262 (row sums) A006152, A114329, A351825.
%K A351823 nonn,tabf
%O A351823 0,4
%A A351823 _Geoffrey Critzer_, Feb 20 2022