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A272519 Number of set partitions of [n] into seven blocks with distinct sizes.

%I A272519 #11 Feb 19 2023 03:17:49
%S A272519 2431106898187968000,8812762505931384000,67144857188048640000,
%T A272519 416298114565901568000,3144312274410635328000,23728992530256389376000,
%U A272519 238675412699786289427200,3207620559498676985664000,16207982672116390803648000,117220515926387332979520000
%N A272519 Number of set partitions of [n] into seven blocks with distinct sizes.
%H A272519 Alois P. Heinz, <a href="/A272519/b272519.txt">Table of n, a(n) for n = 28..1000</a>
%F A272519 a(n) = n! * [x^n*y^7] Product_{n>=1} (1+y*x^n/n!).
%p A272519 b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
%p A272519       or t*(2*i+1-t)/2<n, 0, `if`(n=0, 1, b(n,i-1,t)+
%p A272519       `if`(i>n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
%p A272519     end:
%p A272519 a:= n-> b(n$2, 7):
%p A272519 seq(a(n), n=28..40);
%t A272519 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 A272519 a[n_] := b[n, n, 7];
%t A272519 Table[a[n], {n, 28, 40}] (* _Jean-François Alcover_, May 24 2018, translated from Maple *)
%Y A272519 Column k=7 of A131632.
%K A272519 nonn
%O A272519 28,1
%A A272519 _Alois P. Heinz_, May 01 2016