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A372650
Total sum over all partitions of [n] of the number of minimal blocks.
3
0, 1, 3, 7, 27, 86, 393, 1688, 8291, 44143, 248428, 1480073, 9440049, 63265606, 444309232, 3273807272, 25227429123, 202458174614, 1689474026499, 14636685675142, 131413462612012, 1220636654904548, 11712836883408675, 115956109213769404, 1182944504931376337
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{k=0..n} k * A372762(n,k).
EXAMPLE
a(4) = 27 = 1+1+1+2+2+1+2+2+2+1+2+2+2+2+4: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
MAPLE
b:= proc(n, m, t) option remember; `if`(n=0, t,
add(binomial(n-1, j-1)*b(n-j, min(j, m),
`if`(j<m, 1, `if`(j=m, t+1, t))), j=1..n))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..24);
MATHEMATICA
b[n_, m_, t_] := b[n, m, t] = If[n == 0, t,
Sum[Binomial[n - 1, j - 1]*b[n - j, Min[j, m],
If[j < m, 1, If[j == m, t + 1, t]]], {j, 1, n}]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, May 11 2024, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 08 2024
STATUS
approved