# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a350684
Showing 1-1 of 1

%I A350684 #30 Mar 18 2022 05:21:30
%S A350684 1,0,1,1,1,1,1,2,1,6,3,4,2,16,7,8,14,3,3,1,73,25,26,51,12,12,4,298,91,
%T A350684 92,164,116,56,30,21,4,4,1,1453,390,391,601,676,256,163,147,28,28,7,
%U A350684 7366,1797,1798,2484,3228,1927,897,876,307,307,87,31,31,5,5,1
%N A350684 Number T(n,k) of partitions of [n] such that the sum of elements i contained in block i equals k when blocks are ordered with decreasing largest elements; triangle T(n,k), n>=0, 0<=k<=max(0,A008805(n-1)), read by rows.
%H A350684 Alois P. Heinz, <a href="/A350684/b350684.txt">Rows n = 0..100, flattened</a>
%H A350684 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A350684 Sum_{k=1..max(0,A008805(n-1))} k * T(n,k) = A350683(n).
%F A350684 T(2n,A000217(n)) = A152947(n+1).
%F A350684 T(2n-1,A000217(n)) = 1 for n>=1.
%F A350684 T(n,2) - T(n,1) = 1 for n>=3.
%e A350684 T(4,0) = 6: 432|1, 42|31, 42|3|1, 4|31|2, 4|3|21, 4|3|2|1.
%e A350684 T(4,1) = 3: 432(1), 42(1)|3, 4(1)|3|2.
%e A350684 T(4,2) = 4: 43|(2)1, 43|(2)|1, 4|3(2)1, 4|3(2)|1,
%e A350684 T(4,3) = 2: 43(1)|(2), 4(1)|3(2).
%e A350684 Triangle T(n,k) begins:
%e A350684      1;
%e A350684      0,   1;
%e A350684      1,   1;
%e A350684      1,   1,   2,   1;
%e A350684      6,   3,   4,   2;
%e A350684     16,   7,   8,  14,   3,   3,   1;
%e A350684     73,  25,  26,  51,  12,  12,   4;
%e A350684    298,  91,  92, 164, 116,  56,  30,  21,  4,  4, 1;
%e A350684   1453, 390, 391, 601, 676, 256, 163, 147, 28, 28, 7;
%e A350684   ...
%p A350684 b:= proc(n, m) option remember; expand(`if`(n=0, 1, add(
%p A350684      `if`(n=j, x^j, 1)*b(n-1, max(m, j)), j=1..m+1)))
%p A350684     end:
%p A350684 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
%p A350684 seq(T(n), n=0..10);
%t A350684 b[n_, m_] := b[n, m] = Expand[If[n == 0, 1, Sum[
%t A350684      If[n == j, x^j, 1]*b[n - 1, Max[m, j]], {j, 1, m + 1}]]];
%t A350684 T[n_] := CoefficientList[b[n, 0], x];
%t A350684 Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Mar 18 2022, after _Alois P. Heinz_ *)
%Y A350684 Columns k=0-1 give: A350649, A350650.
%Y A350684 Row sums give A000110.
%Y A350684 Cf. A000217, A008805, A152947, A350647, A350683.
%K A350684 nonn,tabf
%O A350684 0,8
%A A350684 _Alois P. Heinz_, Jan 11 2022

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE