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

0,4

Comments

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

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

Example

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.

Maple

b:= proc(n, i, k) option remember; `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))))
    end:
a:= n-> 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 *)

Crossrefs

Column k=2 of A327869.

Sequence in context: A090460 A071983 A302693 * A215514 A215347 A186247

Adjacent sequences: A327878 A327879 A327880 * A327882 A327883 A327884

Keyword

nonn

Author

Alois P. Heinz, Sep 28 2019

Status

approved