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A266518 Number of ordered partitions of a 2n-set with nondecreasing block sizes and maximal block size equal to n.

%I A266518 #17 Oct 05 2019 20:43:32
%S A266518 1,2,18,200,3290,61992,1480248,39402792,1229123610,42349478600,
%T A266518 1640551617848,69364811821032,3222214209737432,161656803984848200,
%U A266518 8772238289222220600,509677254444910662000,31677425399312755814970,2092539622373193784503240
%N A266518 Number of ordered partitions of a 2n-set with nondecreasing block sizes and maximal block size equal to n.
%H A266518 Alois P. Heinz, <a href="/A266518/b266518.txt">Table of n, a(n) for n = 0..300</a>
%F A266518 a(n) = (2n)! * [x^n] Product_{i=1..n} (i-1)!/(i!-x^i).
%F A266518 a(n) = A262071(2n,n).
%F A266518 a(n) ~ c * 2^(2*n+1/2) * n^n / exp(n), where c = A247551 = 2.529477472079152648... . - _Vaclav Kotesovec_, Jan 02 2016
%F A266518 a(n) = A327801(2n,n). - _Alois P. Heinz_, Sep 26 2019
%p A266518 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A266518        b(n, i-1)+`if`(i>n, 0, binomial(n, i)*b(n-i, i))))
%p A266518     end:
%p A266518 a:= n-> `if`(n=0, 1, b(2*n, n)-b(2*n, n-1)):
%p A266518 seq(a(n), n=0..20);
%t A266518 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, Binomial[n, i]*b[n-i, i]]]]; a[n_] := If[n==0, 1, b[2n, n] - b[2n, n-1]]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 27 2017, translated from Maple *)
%Y A266518 Cf. A262071, A327801, A328156.
%K A266518 nonn
%O A266518 0,2
%A A266518 _Alois P. Heinz_, Dec 30 2015