A009233 Expansion of e.g.f. exp(sinh(x)*x) (even powers only).

1, 2, 16, 246, 5944, 202330, 9099564, 517447126, 36048776656, 3003924569778, 293835907664980, 33232296062419630, 4291773869167401720, 626311538509296801226, 102365694283336181089084, 18595053487766135171539590, 3729223211361742071603266464

Offset

0,2

Comments

Number of ways to choose one element from each block of the partitions of a 2n-set into even blocks (see example). - Enrique Navarrete, Sep 03 2025

Links

Table of n, a(n) for n=0..16.

Formula

a(0) = 1; a(n) = 2 * Sum_{k=1..n} binomial(2*n-1,2*k-1) * k * a(n-k). - Ilya Gutkovskiy, Mar 10 2022

Example

From Enrique Navarrete, Sep 03 2025: (Start)
Seen as even groups of a total of 2*n people, the number of ways to choose one person from each group (block) for n=4 is (number of people in parentheses):
  1 group (8): 1 such group, 8 ways;
  2 groups (6,2): 28 such groups, 336 ways;
  2 groups (4,4): 35 such groups, 560 ways;
  3 groups (4,2,2): 210 such groups, 3360 ways;
  4 groups (2,2,2,2): 105 such groups, 1680 ways, for a total of 5944 ways. (End)

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-2*j)*binomial(n-1, 2*j-1)*2*j, j=1..n/2))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..16);  # Alois P. Heinz, Sep 08 2025

Mathematica

With[{nn=30}, Take[CoefficientList[Series[Exp[Sinh[x]*x], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Jul 31 2020 *)

Prog

(PARI) my(x='x+O('x^40), v=Vec(serlaplace(exp(sinh(x)*x)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

Crossrefs

Cf. A009214, A352254.

Cf. A005046.

Sequence in context: A217814 A217815 A192325 * A188560 A012462 A012457

Adjacent sequences: A009230 A009231 A009232 * A009234 A009235 A009236

Keyword

nonn

Author

R. H. Hardin

Extensions

Extended and signs tested by Olivier Gérard, Mar 15 1997

Previous Mathematica program replaced by Harvey P. Dale, Jul 31 2020

Status

approved