OFFSET
0,4
COMMENTS
Number of ways to choose two elements from each block of the labeled ordered partitions of an n-set into odd blocks (see Example).
If the partitions are not ordered the number of ways is given by A389224.
FORMULA
a(n) ~ n! / ((2 + r*sqrt(1 + r^4/4))*r^n), where r = 1.17122381923483650915059844... is the root of the equation r^2*sinh(r)=2. - Vaclav Kotesovec, Nov 11 2025
EXAMPLE
a(8) = 3360 since the number of ways are:
Sample partition {1,2,3,4,5} {6,7,8}: 112 such ordered partitions, 3360 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 two elements from each block.
a(9) = 45396 since the number of ways are:
Sample partition {1,2,3,4,5,6,7,8,9}: 1 partition, 36 ways.
Sample partition {1,2,3} {4,5,6} {7,8,9}: 1680 such ordered partitions, 45360 ways.
MATHEMATICA
nmax = 20; CoefficientList[Series[2/(2 - x^2*Sinh[x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 11 2025 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Enrique Navarrete, Nov 10 2025
STATUS
approved
