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

1, 0, 5, 42, 569, 9470, 191804, 4534502, 122544881, 3721101192, 125331498349, 4634063018948, 186515332107196, 8114659545679752, 379362605925991692, 18961051425453713478, 1008752282616284996865, 56905048753221935350268, 3392250956149146382053539

Offset: 0

Author

Alois P. Heinz, Jun 01 2023

Keywords

• nonn

Examples

			a(2) = 5: 13|24, 14|2|3, 1|2|34, 1|23|4, 12|3|4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• Wikipedia, Partition of a set

Crossrefs

Cf. A124424, A363434, A363451.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1, expand(x*
      add(g(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= (n, k)-> coeff(g(n), x, k):
b:= proc(g, u) option remember;
      add(S(g, k)*S(u, k)*k!, k=0..min(g, u))
    end:
T:= proc(n, k) option remember; local g, u; g:= floor(n/2); u:= ceil(n/2);
      add(add(add(binomial(g, i)*S(i, h)*binomial(u, j)*
      S(j, k-h)*b(g-i, u-j), j=k-h..u), i=h..g), h=0..k)
    end:
a:= n-> T(2*n, n):
seq(a(n), n=0..18);

Mathematica

b[g_, u_] := b[g, u] = Sum[StirlingS2[g, k]*StirlingS2[u, k]*k!, {k, 0, Min[g, u]}];
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}]];
a[n_] := T[2n, n];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 20 2023, after Alois P. Heinz in A124424 *)

Formula

a(n) = A124424(2n,n).

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