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A367820
Number of partitions of [2n] that have at most one block contained in [n].
2
1, 2, 13, 153, 2744, 68303, 2224417, 90995838, 4538437039, 269755223485, 18766884323562, 1506040068195721, 137740473851280141, 14212098473767962472, 1640078704487165930485, 210103319793655159244093, 29684467774817808296383256, 4598958815992575305097910699
OFFSET
0,2
LINKS
FORMULA
a(n) = A113547(2n+1,n+1) = A362925(2n,n).
a(n) = Sum_{j=0..n} (j+1)^n * Stirling2(n,j).
a(n) mod 2 = A011655(n+2).
EXAMPLE
a(2) = 13: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3.
MAPLE
b:= proc(n) option remember; expand(`if`(n=0, 1,
x*add(b(n-j)*binomial(n-1, j-1), j=1..n)))
end:
a:= n-> add(coeff(b(n), x, j)*(j+1)^n, j=0..n):
seq(a(n), n=0..21);
MATHEMATICA
A367820[n_]:=Sum[StirlingS2[n, j](j+1)^n, {j, 0, n}];
Array[A367820, 25, 0] (* Paolo Xausa, Dec 04 2023 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Dec 01 2023
STATUS
approved