A387547 Number of partitions of a 2n-set into even blocks of size > 2.

1, 0, 1, 1, 36, 211, 6733, 109110, 3935179, 120852049, 5443425654, 255558333097, 14770120409245, 943648512760656, 69287778033581461, 5659960611372895981, 516263085381593167788, 51978891572002399746295, 5755459912529835391400569, 696667046057392012794065454

Offset

0,5

Comments

a(n) is the number of ways to form even size groups from a total of 2*n people, where each group has at least 4 people.

Links

Table of n, a(n) for n=0..19.

Formula

a(n) = (2*n)! * [x^(2*n)] exp(cosh(x) - x^2/2 - 1).

a(n) = Sum_{k=2..n} binomial(2*n-1, 2*k-1)*a(n-k) for n > 0, a(0)=1.

Example

a(3) = 1 since for 6 people we can form a single group if a group has to have at least 4 people.
a(6) = 6733 since for 12 people the number of ways are (number of people in parentheses):
  1 group (12): 1 way;
  2 groups (8,4): 495 ways;
  2 groups (6,6): 462 ways;
  3 groups (4,4,4): 5775 ways.

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-2*j)*binomial(n-1, 2*j-1), j=2..n/2))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..19);  # Alois P. Heinz, Sep 01 2025

Mathematica

a[n_]:=(2*n)!*SeriesCoefficient[Exp[Cosh[x]-x^2/2-1], {x, 0, 2n}]; Array[a, 20, 0] (* Stefano Spezia, Sep 02 2025 *)

Crossrefs

Cf. A000110, A005046.

Sequence in context: A120465 A361209 A144523 * A232677 A232670 A365308

Adjacent sequences: A387544 A387545 A387546 * A387548 A387549 A387550

Keyword

nonn,easy

Author

Enrique Navarrete, Sep 01 2025

Status

approved