A367820 Number of partitions of [2n] that have at most one block contained in [n].

1, 2, 13, 153, 2744, 68303, 2224417, 90995838, 4538437039, 269755223485, 18766884323562, 1506040068195721, 137740473851280141, 14212098473767962472, 1640078704487165930485, 210103319793655159244093, 29684467774817808296383256, 4598958815992575305097910699

Offset: 0

Author

Alois P. Heinz, Dec 01 2023

Keywords

• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..288

Crossrefs

Cf. A008277, A048993, A011655, A108459, A113547, A362925.

Programs

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 *)

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).