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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..300

Wikipedia, Partition of a set

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

Example

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

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 *)

Crossrefs

Cf. A124424, A363434, A363451.

Sequence in context: A239997 A102244 A323313 * A337593 A226987 A336605

Adjacent sequences: A363432 A363433 A363434 * A363436 A363437 A363438

Keyword

nonn

Author

Alois P. Heinz, Jun 01 2023

Status

approved