A327876 - OEIS

%I #13 Dec 18 2020 04:02:09

%S 0,1,0,3,4,5,66,112,456,765,15070,31856,150756,663962,1943046,

%T 44316105,127348864,661449549,3220447446,20913769072,68889553260,

%U 2403704171190,7894983374674,51012052828947,270186003789312,1836634462055350,13402212376611266

%N Number of set partitions of [n] with distinct block sizes and one of the block sizes is 1.

%C Sum of multinomials M(n; lambda), where lambda ranges over all integer partitions of n into distinct parts and one part is 1.

%H Alois P. Heinz, <a href="/A327876/b327876.txt">Table of n, a(n) for n = 0..697</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%F a(1) = 1: 1.

%F a(2) = 0.

%F a(3) = 3: 12|3, 13|2, 1|23.

%F a(4) = 4: 123|4, 124|3, 134|2, 1|234.

%F a(5) = 5: 1234|5, 1235|4, 1245|3, 1345|2, 1|2345.

%F a(6) = 66: 12345|6, 12346|5, 12356|4, 123|45|6, 123|46|5, 123|4|56, 12456|3, 124|35|6, 124|36|5, 124|3|56, 125|34|6, 12|345|6, 126|34|5, 12|346|5, 125|36|4, 125|3|46, 126|35|4, 12|356|4, 126|3|45, 12|3|456, 13456|2, 134|25|6, 134|26|5, 134|2|56, 135|24|6, 13|245|6, 136|24|5, 13|246|5, 135|26|4, 135|2|46, 136|25|4, 13|256|4, 136|2|45, 13|2|456, 145|23|6, 14|235|6, 146|23|5, 14|236|5, 15|234|6, 1|23456, 16|234|5, 1|234|56, 156|23|4, 15|236|4, 16|235|4, 1|235|46, 1|236|45, 1|23|456, 145|26|3, 145|2|36, 146|25|3, 14|256|3, 146|2|35, 14|2|356, 156|24|3, 15|246|3, 16|245|3, 1|245|36, 1|246|35, 1|24|356, 156|2|34, 15|2|346, 1|256|34, 1|25|346, 16|2|345, 1|26|345.

%p b:= proc(n, i, k) option remember; `if`(i*(i+1)/2<n, 0,

%p      `if`(n=0, 1, `if`(i<2, 0, b(n, i-1, `if`(i=k, 0, k)))+

%p      `if`(i=k, 0, b(n-i, min(n-i, i-1), k)*binomial(n, i))))

%p     end:

%p a:= n-> b(n$2, 0)-b(n$2, 1):

%p seq(a(n), n=0..29);

%t 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]]]];

%t a[n_] := b[n, n, 0] - b[n, n, 1];

%t a /@ Range[0, 29] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)

%Y Column k=1 of A327869.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Sep 28 2019