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A272516 Number of set partitions of [n] into four blocks with distinct sizes.

%I A272516 #7 Dec 11 2020 08:18:30
%S A272516 12600,27720,138600,643500,4408404,12687675,60780720,238299880,
%T A272516 1295666424,4208874756,18840460800,72351683460,361100656224,
%U A272516 1228553894491,5370616442928,20605640103400,97659853077800,342942099783075,1479570975628200,5678915129142255
%N A272516 Number of set partitions of [n] into four blocks with distinct sizes.
%H A272516 Alois P. Heinz, <a href="/A272516/b272516.txt">Table of n, a(n) for n = 10..1000</a>
%F A272516 a(n) = n! * [x^n*y^4] Product_{n>=1} (1+y*x^n/n!).
%p A272516 b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
%p A272516       or t*(2*i+1-t)/2<n, 0, `if`(n=0, 1, b(n,i-1,t)+
%p A272516       `if`(i>n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
%p A272516     end:
%p A272516 a:= n-> b(n$2, 4):
%p A272516 seq(a(n), n=10..40);
%t A272516 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 A272516 a[n_] := b[n, n, 4];
%t A272516 a /@ Range[10, 40] (* _Jean-François Alcover_, Dec 11 2020, after _Alois P. Heinz_ *)
%Y A272516 Column k=4 of A131632.
%K A272516 nonn
%O A272516 10,1
%A A272516 _Alois P. Heinz_, May 01 2016