A368204 - OEIS

%I #17 Jun 10 2024 10:02:50

%S 1,1,0,2,2,2,29,56,191,380,5097,14288,74359,283884,1106529,13588409,

%T 53640963,350573155,1867738775,10770352150,50050737949,744605446778,

%U 3615378756421,29368052533243,195027586980839,1442227919200245,8964685271444243,61478734886319324

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

%H Alois P. Heinz, <a href="/A368204/b368204.txt">Table of n, a(n) for n = 0..677</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%F a(n) = A124327(n,n).

%e a(0) = 1: the empty partition.

%e a(1) = 1: 1.

%e a(2) = 0.

%e a(3) = 2: 13|2, 1|23.

%e a(4) = 2: 124|3, 12|34.

%e a(5) = 2: 1235|4, 123|45.

%e 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 b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),

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

%p       t*b(n, i+1, t, m))+ b(n-i, i+1, t+1, m)))

%p     end:

%p a:= n-> b(n, 1, 0, n):

%p seq(a(n), n=0..42);

%t b[n_, i_, t_, m_] := b[n, i, t, m] = If[n == 0, t^(m - i + 1),

%t    If[(i + m)*(m + 1 - i)/2 < n || i > n, 0, If[t == 0, 0,

%t    t*b[n, i + 1, t, m]] + b[n - i, i + 1, t + 1, m]]];

%t a[n_] := If[n == 0, 1, b[n, 1, 0, n]];

%t Table[a[n], {n, 0, 42}] (* _Jean-François Alcover_, Jun 10 2024, after _Alois P. Heinz_ *)

%Y Main diagonal of A124327.

%Y Cf. A365441, A368246.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Dec 16 2023