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

%I A327881 #10 Dec 18 2020 04:02:16
%S A327881 0,0,1,3,0,10,75,126,196,1548,15525,39820,161106,358722,3705884,
%T A327881 46623045,142988280,768721504,3560215293,12250746432,144581799790,
%U A327881 2542575063630,8955836934660,55657973021431,319349051391228,1983548989621200,7898257536096850
%N A327881 Number of set partitions of [n] with distinct block sizes and one of the block sizes is 2.
%C A327881 Sum of multinomials M(n; lambda), where lambda ranges over all integer partitions of n into distinct parts and one part is 2.
%H A327881 Alois P. Heinz, <a href="/A327881/b327881.txt">Table of n, a(n) for n = 0..697</a>
%H A327881 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%H A327881 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A327881 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A327881 a(2) = 1: 12.
%e A327881 a(3) = 3: 12|3, 13|2, 1|23.
%e A327881 a(4) = 0.
%e A327881 a(5) = 10: 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234.
%p A327881 b:= proc(n, i, k) option remember; `if`(i*(i+1)/2<n, 0,
%p A327881      `if`(n=0, 1, `if`(i<2, 0, b(n, i-1, `if`(i=k, 0, k)))+
%p A327881      `if`(i=k, 0, b(n-i, min(n-i, i-1), k)*binomial(n, i))))
%p A327881     end:
%p A327881 a:= n-> b(n$2, 0)-b(n$2, 2):
%p A327881 seq(a(n), n=0..29);
%t A327881 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 A327881 a[n_] := b[n, n, 0] - b[n, n, 2];
%t A327881 a /@ Range[0, 29] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A327881 Column k=2 of A327869.
%K A327881 nonn
%O A327881 0,4
%A A327881 _Alois P. Heinz_, Sep 28 2019