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%I #22 Jul 16 2023 16:41:55
%S 1,0,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,3,2,1,0,0,1,3,7,2,1,0,0,1,7,14,
%T 13,3,1,0,0,1,7,35,26,22,3,1,0,0,1,15,70,113,66,34,4,1,0,0,1,15,155,
%U 226,311,102,50,4,1,0,0,1,31,310,833,933,719,200,70,5,1
%N Number T(n,k) of set partitions of [n] into k blocks such that each element is contained in a block whose index parity coincides with the parity of the element; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C All odd elements are in blocks with an odd index and all even elements are in blocks with an even index.
%H Alois P. Heinz, <a href="/A274537/b274537.txt">Rows n = 0..140, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F Sum_{k=0..n} k * T(n,k) = A364267(n). - _Alois P. Heinz_, Jul 16 2023
%e T(6,2) = 1: 135|246.
%e T(6,3) = 3: 13|246|5, 15|246|3, 1|246|35.
%e T(6,4) = 7: 13|24|5|6, 15|24|3|6, 1|24|35|6, 15|26|3|4, 15|2|3|46, 1|26|35|4, 1|2|35|46.
%e T(6,5) = 2: 1|26|3|4|5, 1|2|3|46|5.
%e T(6,6) = 1: 1|2|3|4|5|6.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 0, 1;
%e 0, 0, 1, 1;
%e 0, 0, 1, 1, 1;
%e 0, 0, 1, 3, 2, 1;
%e 0, 0, 1, 3, 7, 2, 1;
%e 0, 0, 1, 7, 14, 13, 3, 1;
%e 0, 0, 1, 7, 35, 26, 22, 3, 1;
%e 0, 0, 1, 15, 70, 113, 66, 34, 4, 1;
%e 0, 0, 1, 15, 155, 226, 311, 102, 50, 4, 1;
%e ...
%p b:= proc(n, m, t) option remember; `if`(n=0, x^m, add(
%p `if`(irem(j, 2)=t, b(n-1, max(m, j), 1-t), 0), j=1..m+1))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0, 1)):
%p seq(T(n), n=0..12);
%t b[n_, m_, t_] := b[n, m, t] = If[n==0, x^m, Sum[If[Mod[j, 2]==t, b[n-1, Max[m, j], 1-t], 0], {j, 1, m+1}]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0, 1]]; Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 18 2016, after _Alois P. Heinz_ *)
%Y Row sums give A274538.
%Y Columns k=0-10 give: A000007, A000007(n-1), A000012(n-2), A052551(n-3), A274868, A274869, A274870, A274871, A274872, A274873, A274874.
%Y T(2n,n) gives A274875.
%Y Main diagonal and lower diagonals give: A000012, A004526, A002623(n-2) or A173196.
%Y Cf. A364267.
%K nonn,tabl
%O 0,19
%A _Alois P. Heinz_, Jun 27 2016
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