A355144 - OEIS

%I #22 Jun 25 2022 12:54:40

%S 1,1,2,4,1,10,5,26,26,76,117,10,232,540,105,764,2445,931,2620,11338,

%T 6909,280,9496,53033,48546,4900,35696,253826,324753,64295,140152,

%U 1235115,2131855,691075,15400,568504,6142878,13792779,6739876,400400,2390480,31126539,88890880,61274213,7217210

%N Number T(n,k) of partitions of [n] having exactly k blocks of size at least three; triangle T(n,k), n>=0, 0<=k<=floor(n/3), read by rows.

%H Alois P. Heinz, <a href="/A355144/b355144.txt">Rows n = 0..250, flattened</a>

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

%F Sum_{k=1..n} k * T(n,k) = A288785(n).

%e T(4,1) = 5: 1234, 123|4, 124|3, 134|2, 1|234.

%e T(6,2) = 10: 123|456, 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.

%e Triangle T(n,k) begins:

%e        1;

%e        1;

%e        2;

%e        4,       1;

%e       10,       5;

%e       26,      26;

%e       76,     117,      10;

%e      232,     540,     105;

%e      764,    2445,     931;

%e     2620,   11338,    6909,    280;

%e     9496,   53033,   48546,   4900;

%e    35696,  253826,  324753,  64295;

%e   140152, 1235115, 2131855, 691075, 15400;

%e   ...

%p b:= proc(n) option remember; expand(`if`(n=0, 1, add(

%p      `if`(i>2, x, 1)*binomial(n-1, i-1)*b(n-i), i=1..n)))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):

%p seq(T(n), n=0..14);  # _Alois P. Heinz_, Jun 20 2022

%t b[n_] := b[n] = Expand[If[n == 0, 1, Sum[If[i > 2, x, 1]*

%t      Binomial[n - 1, i - 1]*b[n - i], {i, 1, n}]]];

%t T[n_] := CoefficientList[b[n], x];

%t Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Jun 25 2022, after _Alois P. Heinz_ *)

%Y Column k=0 gives A000085.

%Y Row sums give A000110.

%Y T(3n,n) gives A025035.

%Y Cf. A048993, A124324, A124503, A288785.

%K nonn,tabf

%O 0,3

%A _Alois P. Heinz_, Jun 20 2022