OFFSET
0,4
COMMENTS
Number of ways to choose an even number of elements from each block of the partitions of an n-set into odd blocks (see Example).
EXAMPLE
a(7) = 63 since from the sample partition {1,2,3,4,5,6,7} we can choose 2 elements in 21 ways, 4 elements in 35 ways, and 6 elements in 7 ways.
Note that for other partitions into odd blocks such as {1,2,3,4,5} {6} {7} it is not possible to choose an even number of elements from each block.
a(8) = 2520 since from the sample partition {1,2,3,4,5} {6,7,8} (56 such partitions) we can choose:
i) 2 elements from each block in 1680 ways;
ii) 4 elements from the first block and 2 elements from the second block in 840 ways.
Note that for other partitions into odd blocks such as {1,2,3} {4,5,6} {7} {8} it is not possible to choose an even number of elements from each block.
MATHEMATICA
With[{nn=30}, CoefficientList[Series[Exp[Sinh[x](Cosh[x]-1)], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Nov 18 2025 *)
PROG
(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(sinh(x) * (cosh(x)-1)))) \\ Michel Marcus, Oct 03 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Enrique Navarrete, Oct 03 2025
STATUS
approved