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A363435 Number of partitions of [2n] having exactly n blocks with all elements of the same parity.

%I A363435 #21 Oct 21 2023 03:26:58
%S A363435 1,0,5,42,569,9470,191804,4534502,122544881,3721101192,125331498349,
%T A363435 4634063018948,186515332107196,8114659545679752,379362605925991692,
%U A363435 18961051425453713478,1008752282616284996865,56905048753221935350268,3392250956149146382053539
%N A363435 Number of partitions of [2n] having exactly n blocks with all elements of the same parity.
%H A363435 Alois P. Heinz, <a href="/A363435/b363435.txt">Table of n, a(n) for n = 0..300</a>
%H A363435 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A363435 a(n) = A124424(2n,n).
%F A363435 Conjecture: Limit_{n->oo} (a(n)/n!)^(1/n) = A238258 = -2 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 3.0882773... - _Vaclav Kotesovec_, Oct 21 2023
%e A363435 a(2) = 5: 13|24, 14|2|3, 1|2|34, 1|23|4, 12|3|4.
%p A363435 g:= proc(n) option remember; `if`(n=0, 1, expand(x*
%p A363435       add(g(n-j)*binomial(n-1, j-1), j=1..n)))
%p A363435     end:
%p A363435 S:= (n, k)-> coeff(g(n), x, k):
%p A363435 b:= proc(g, u) option remember;
%p A363435       add(S(g, k)*S(u, k)*k!, k=0..min(g, u))
%p A363435     end:
%p A363435 T:= proc(n, k) option remember; local g, u; g:= floor(n/2); u:= ceil(n/2);
%p A363435       add(add(add(binomial(g, i)*S(i, h)*binomial(u, j)*
%p A363435       S(j, k-h)*b(g-i, u-j), j=k-h..u), i=h..g), h=0..k)
%p A363435     end:
%p A363435 a:= n-> T(2*n, n):
%p A363435 seq(a(n), n=0..18);
%t A363435 b[g_, u_] := b[g, u] = Sum[StirlingS2[g, k]*StirlingS2[u, k]*k!, {k, 0, Min[g, u]}];
%t A363435 T[n_, k_] := Module[{g, u}, g = Floor[n/2]; u = Ceiling[n/2]; Sum[Sum[Sum[ Binomial[g, i]*StirlingS2[i, h]*Binomial[u, j]*StirlingS2[j, k - h]*b[g - i, u - j], {j, k - h, u}], {i, h, g}], {h, 0, k}]];
%t A363435 a[n_] := T[2n, n];
%t A363435 Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Oct 20 2023, after _Alois P. Heinz_ in A124424 *)
%Y A363435 Cf. A124424, A363434, A363451.
%K A363435 nonn
%O A363435 0,3
%A A363435 _Alois P. Heinz_, Jun 01 2023