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A368204 Number of partitions of [n] whose block minima sum to n.

%I A368204 #17 Jun 10 2024 10:02:50
%S A368204 1,1,0,2,2,2,29,56,191,380,5097,14288,74359,283884,1106529,13588409,
%T A368204 53640963,350573155,1867738775,10770352150,50050737949,744605446778,
%U A368204 3615378756421,29368052533243,195027586980839,1442227919200245,8964685271444243,61478734886319324
%N A368204 Number of partitions of [n] whose block minima sum to n.
%H A368204 Alois P. Heinz, <a href="/A368204/b368204.txt">Table of n, a(n) for n = 0..677</a>
%H A368204 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A368204 a(n) = A124327(n,n).
%e A368204 a(0) = 1: the empty partition.
%e A368204 a(1) = 1: 1.
%e A368204 a(2) = 0.
%e A368204 a(3) = 2: 13|2, 1|23.
%e A368204 a(4) = 2: 124|3, 12|34.
%e A368204 a(5) = 2: 1235|4, 123|45.
%e A368204 a(6) = 29: 12346|5, 1234|56, 1456|2|3, 145|26|3, 145|2|36, 146|25|3, 14|256|3, 14|25|36, 146|2|35, 14|26|35, 14|2|356, 156|24|3, 15|246|3, 15|24|36, 16|245|3, 1|2456|3, 1|245|36, 16|24|35, 1|246|35, 1|24|356, 156|2|34, 15|26|34, 15|2|346, 16|25|34, 1|256|34, 1|25|346, 16|2|345, 1|26|345, 1|2|3456.
%p A368204 b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
%p A368204      `if`((i+m)*(m+1-i)/2<n or i>n, 0, `if`(t=0, 0,
%p A368204       t*b(n, i+1, t, m))+ b(n-i, i+1, t+1, m)))
%p A368204     end:
%p A368204 a:= n-> b(n, 1, 0, n):
%p A368204 seq(a(n), n=0..42);
%t A368204 b[n_, i_, t_, m_] := b[n, i, t, m] = If[n == 0, t^(m - i + 1),
%t A368204    If[(i + m)*(m + 1 - i)/2 < n || i > n, 0, If[t == 0, 0,
%t A368204    t*b[n, i + 1, t, m]] + b[n - i, i + 1, t + 1, m]]];
%t A368204 a[n_] := If[n == 0, 1, b[n, 1, 0, n]];
%t A368204 Table[a[n], {n, 0, 42}] (* _Jean-François Alcover_, Jun 10 2024, after _Alois P. Heinz_ *)
%Y A368204 Main diagonal of A124327.
%Y A368204 Cf. A365441, A368246.
%K A368204 nonn
%O A368204 0,4
%A A368204 _Alois P. Heinz_, Dec 16 2023