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A272515 Number of set partitions of [n] into three blocks with distinct sizes.

%I A272515 #10 Dec 11 2020 08:25:19
%S A272515 60,105,448,2016,4980,15675,61644,155155,482573,1733550,4549808,
%T A272515 13890360,48104628,128949675,392009140,1322692581,3607864403,
%U A272515 10929721440,36245555284,100109572875,302709337515,990788537700,2763564406113,8344789976616,27039048750600
%N A272515 Number of set partitions of [n] into three blocks with distinct sizes.
%H A272515 Alois P. Heinz, <a href="/A272515/b272515.txt">Table of n, a(n) for n = 6..1000</a>
%F A272515 a(n) = n! * [x^n*y^3] Product_{n>=1} (1+y*x^n/n!).
%F A272515 Conjecture: a(n) ~ 3^n / 6. - _Vaclav Kotesovec_, Dec 11 2020
%p A272515 b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
%p A272515       or t*(2*i+1-t)/2<n, 0, `if`(n=0, 1, b(n,i-1,t)+
%p A272515       `if`(i>n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
%p A272515     end:
%p A272515 a:= n-> b(n$2, 3):
%p A272515 seq(a(n), n=6..40);
%t A272515 b[n_, i_, t_] := b[n, i, t] = If[t > i || t(t+1)/2 > n || t(2i+1-t)/2 < n, 0, If[n == 0, 1, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t - 1]* Binomial[n, i]]]];
%t A272515 a[n_] := b[n, n, 3];
%t A272515 a /@ Range[6, 40] (* _Jean-François Alcover_, Dec 11 2020, after _Alois P. Heinz_ *)
%Y A272515 Column k=3 of A131632.
%K A272515 nonn
%O A272515 6,1
%A A272515 _Alois P. Heinz_, May 01 2016