A327881 Number of set partitions of [n] with distinct block sizes and one of the block sizes is 2.

0, 0, 1, 3, 0, 10, 75, 126, 196, 1548, 15525, 39820, 161106, 358722, 3705884, 46623045, 142988280, 768721504, 3560215293, 12250746432, 144581799790, 2542575063630, 8955836934660, 55657973021431, 319349051391228, 1983548989621200, 7898257536096850

Offset: 0

Author

Alois P. Heinz, Sep 28 2019

Keywords

• nonn

Comments

Sum of multinomials M(n; lambda), where lambda ranges over all integer partitions of n into distinct parts and one part is 2.

Examples

			a(2) = 1: 12.
a(3) = 3: 12|3, 13|2, 1|23.
a(4) = 0.
a(5) = 10: 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..697
• Wikipedia, Multinomial coefficients
• Wikipedia, Partition (number theory)
• Wikipedia, Partition of a set

Crossrefs

Column k=2 of A327869.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 b(n$2, 0)-b(n$2, 2):
seq(a(n), n=0..29);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n == 0, 1, If[i < 2, 0, b[n, i - 1, If[i == k, 0, k]]] + If[i == k, 0, b[n - i, Min[n - i, i - 1], k] Binomial[n, i]]]];
a[n_] := b[n, n, 0] - b[n, n, 2];
a /@ Range[0, 29] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)