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A274581 Number T(n,k) of set partitions of [n] with alternating parity of elements and exactly k blocks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A274581 #28 Jun 09 2023 14:41:13
%S A274581 1,0,1,0,1,1,0,1,2,1,0,1,3,3,1,0,1,5,7,4,1,0,1,7,14,12,5,1,0,1,11,30,
%T A274581 33,19,6,1,0,1,15,57,84,62,27,7,1,0,1,23,119,222,204,108,37,8,1,0,1,
%U A274581 31,224,545,627,409,169,48,9,1,0,1,47,460,1425,2006,1558,763,254,61,10,1
%N A274581 Number T(n,k) of set partitions of [n] with alternating parity of elements and exactly k blocks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A274581 Alois P. Heinz, <a href="/A274581/b274581.txt">Rows n = 0..35, flattened</a>
%H A274581 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A274581 Sum_{k=0..n} k * T(n,k) = A305823(n).
%e A274581 T(5,1) = 1: 12345.
%e A274581 T(5,2) = 5: 1234|5, 123|45, 12|345, 145|23, 1|2345.
%e A274581 T(5,3) = 7: 123|4|5, 12|34|5, 12|3|45, 1|234|5, 145|2|3, 1|2|345, 1|23|45.
%e A274581 T(5,4) = 4: 12|3|4|5, 1|23|4|5, 1|2|34|5, 1|2|3|45.
%e A274581 T(5,5) = 1: 1|2|3|4|5.
%e A274581 Triangle T(n,k) begins:
%e A274581   1;
%e A274581   0, 1;
%e A274581   0, 1,  1;
%e A274581   0, 1,  2,   1;
%e A274581   0, 1,  3,   3,   1;
%e A274581   0, 1,  5,   7,   4,   1;
%e A274581   0, 1,  7,  14,  12,   5,   1;
%e A274581   0, 1, 11,  30,  33,  19,   6,   1;
%e A274581   0, 1, 15,  57,  84,  62,  27,   7,  1;
%e A274581   0, 1, 23, 119, 222, 204, 108,  37,  8, 1;
%e A274581   0, 1, 31, 224, 545, 627, 409, 169, 48, 9, 1;
%e A274581   ...
%p A274581 b:= proc(l, i, t) option remember; `if`(l=[], x,
%p A274581      `if`(l[1]=t, 0, expand(x*b(subsop(1=[][], l), 1, 1-t)
%p A274581        ))+add(`if`(l[j]=t, 0, b(subsop(j=[][], l), j, 1-t)
%p A274581        ), j=i..nops(l)))
%p A274581     end:
%p A274581 T:= n-> `if`(n=0, 1, (p-> seq(coeff(p, x, j), j=0..n))(
%p A274581          b([seq(irem(i, 2), i=2..n)], 1$2))):
%p A274581 seq(T(n), n=0..12);
%t A274581 b[l_, i_, t_] := b[l, i, t] = If[l == {}, x, If[l[[1]] == t, 0, Expand[x*b[Rest[l], 1, 1 - t]]] + Sum[If[l[[j]] == t, 0, b[Delete[l, j], j, 1 - t]], {j, i, Length[l]}]];
%t A274581 T[n_] := If[n==0, {1}, Function[p, Table[Coefficient[p, x, j], {j, 0, n}]][ b[Table[Mod[i, 2], {i, 2, n}], 1, 1]]];
%t A274581 Flatten[Table[T[n], {n, 0, 12}]] (* _Jean-François Alcover_, May 27 2018, from Maple *)
%Y A274581 Columns k=0-10 give: A000007, A057427, A052955(n-2) for n>1, A305777, A305778, A305779, A305780, A305781, A305782, A305783, A305784.
%Y A274581 Diagonals include A000012, A001477, A077043.
%Y A274581 Row sums give A274547.
%Y A274581 T(n,ceiling(n/2)) gives A305785.
%Y A274581 Cf. A124419, A274310 (parities alternate within blocks), A305823.
%K A274581 nonn,tabl
%O A274581 0,9
%A A274581 _Alois P. Heinz_, Jun 29 2016