%I #11 Feb 19 2023 03:17:49

%S 2431106898187968000,8812762505931384000,67144857188048640000,

%T 416298114565901568000,3144312274410635328000,23728992530256389376000,

%U 238675412699786289427200,3207620559498676985664000,16207982672116390803648000,117220515926387332979520000

%N Number of set partitions of [n] into seven blocks with distinct sizes.

%H Alois P. Heinz, <a href="/A272519/b272519.txt">Table of n, a(n) for n = 28..1000</a>

%F a(n) = n! * [x^n*y^7] Product_{n>=1} (1+y*x^n/n!).

%p b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n

%p or t*(2*i+1-t)/2<n, 0, `if`(n=0, 1, b(n,i-1,t)+

%p `if`(i>n, 0, b(n-i, i-1, t-1)*binomial(n,i))))

%p end:

%p a:= n-> b(n$2, 7):

%p seq(a(n), n=28..40);

%t b[n_, i_, t_] := b[n, i, t] = If[t > i || t*(t + 1)/2 > n || t*(2*i + 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 a[n_] := b[n, n, 7];

%t Table[a[n], {n, 28, 40}] (* _Jean-François Alcover_, May 24 2018, translated from Maple *)

%Y Column k=7 of A131632.

%K nonn

%O 28,1

%A _Alois P. Heinz_, May 01 2016