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A363472
Total number of blocks in all partitions of [n] where each block has at least one odd element and at least one even element.
1
0, 0, 1, 1, 5, 13, 55, 193, 941, 4081, 22351, 113761, 694565, 4030153, 27107095, 175738753, 1289775821, 9209233921, 73147903471, 568928274961, 4857161139365, 40796613003433, 372190216061335, 3352314486348433, 32518958606637101, 312271731474218881
OFFSET
0,5
COMMENTS
All positive terms are odd.
LINKS
FORMULA
a(n) = Sum_{k=0..floor(n/2)} k * Stirling2(floor(n/2),k) * Stirling2(ceiling(n/2),k) * k!.
EXAMPLE
a(2) = 1: 12.
a(3) = 1: 123.
a(4) = 5 = 1 + 2 + 2: 1234, 12|34, 14|23.
a(5) = 13 = 1 + 2 + 2 + 2 + 2 + 2 + 2: 12345, 123|45, 125|34, 12|345, 134|25, 145|23, 14|235.
MAPLE
a:= n-> (h-> add(k*Stirling2(h, k)*Stirling2(n-h, k)*k!, k=0..h))(floor(n/2)):
seq(a(n), n=0..30);
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 05 2023
STATUS
approved