cp's OEIS Frontend

A355144 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.

%I A355144 #22 Jun 25 2022 12:54:40
%S A355144 1,1,2,4,1,10,5,26,26,76,117,10,232,540,105,764,2445,931,2620,11338,
%T A355144 6909,280,9496,53033,48546,4900,35696,253826,324753,64295,140152,
%U A355144 1235115,2131855,691075,15400,568504,6142878,13792779,6739876,400400,2390480,31126539,88890880,61274213,7217210
%N A355144 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 A355144 Alois P. Heinz, <a href="/A355144/b355144.txt">Rows n = 0..250, flattened</a>
%H A355144 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A355144 Sum_{k=1..n} k * T(n,k) = A288785(n).
%e A355144 T(4,1) = 5: 1234, 123|4, 124|3, 134|2, 1|234.
%e A355144 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 A355144 Triangle T(n,k) begins:
%e A355144        1;
%e A355144        1;
%e A355144        2;
%e A355144        4,       1;
%e A355144       10,       5;
%e A355144       26,      26;
%e A355144       76,     117,      10;
%e A355144      232,     540,     105;
%e A355144      764,    2445,     931;
%e A355144     2620,   11338,    6909,    280;
%e A355144     9496,   53033,   48546,   4900;
%e A355144    35696,  253826,  324753,  64295;
%e A355144   140152, 1235115, 2131855, 691075, 15400;
%e A355144   ...
%p A355144 b:= proc(n) option remember; expand(`if`(n=0, 1, add(
%p A355144      `if`(i>2, x, 1)*binomial(n-1, i-1)*b(n-i), i=1..n)))
%p A355144     end:
%p A355144 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
%p A355144 seq(T(n), n=0..14);  # _Alois P. Heinz_, Jun 20 2022
%t A355144 b[n_] := b[n] = Expand[If[n == 0, 1, Sum[If[i > 2, x, 1]*
%t A355144      Binomial[n - 1, i - 1]*b[n - i], {i, 1, n}]]];
%t A355144 T[n_] := CoefficientList[b[n], x];
%t A355144 Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Jun 25 2022, after _Alois P. Heinz_ *)
%Y A355144 Column k=0 gives A000085.
%Y A355144 Row sums give A000110.
%Y A355144 T(3n,n) gives A025035.
%Y A355144 Cf. A048993, A124324, A124503, A288785.
%K A355144 nonn,tabf
%O A355144 0,3
%A A355144 _Alois P. Heinz_, Jun 20 2022