%I #8 Nov 18 2025 15:28:51

%S 1,0,0,3,0,15,90,63,2520,7815,51030,624723,2164140,34058115,242612370,

%T 1902717183,26138745360,181173938415,2529384841710,26353678822443,

%U 274577416776180,4014332266670715,41374637093824650,612050514318556503,7980832911566136600,104885825773367199015

%N Expansion of e.g.f. exp(sinh(x) * (cosh(x)-1)).

%C 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).

%e 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.

%e 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.

%e a(8) = 2520 since from the sample partition {1,2,3,4,5} {6,7,8} (56 such partitions) we can choose:

%e i) 2 elements from each block in 1680 ways;

%e ii) 4 elements from the first block and 2 elements from the second block in 840 ways.

%e 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.

%t With[{nn=30},CoefficientList[Series[Exp[Sinh[x](Cosh[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Nov 18 2025 *)

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(sinh(x) * (cosh(x)-1)))) \\ _Michel Marcus_, Oct 03 2025

%Y Cf. A003724, A009229, A387019, A389224.

%K nonn

%O 0,4

%A _Enrique Navarrete_, Oct 03 2025