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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.
13
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, 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, 31, 224, 545, 627, 409, 169, 48, 9, 1, 0, 1, 47, 460, 1425, 2006, 1558, 763, 254, 61, 10, 1
OFFSET
0,9
LINKS
FORMULA
Sum_{k=0..n} k * T(n,k) = A305823(n).
EXAMPLE
T(5,1) = 1: 12345.
T(5,2) = 5: 1234|5, 123|45, 12|345, 145|23, 1|2345.
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.
T(5,4) = 4: 12|3|4|5, 1|23|4|5, 1|2|34|5, 1|2|3|45.
T(5,5) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
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, 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, 31, 224, 545, 627, 409, 169, 48, 9, 1;
...
MAPLE
b:= proc(l, i, t) option remember; `if`(l=[], x,
`if`(l[1]=t, 0, expand(x*b(subsop(1=[][], l), 1, 1-t)
))+add(`if`(l[j]=t, 0, b(subsop(j=[][], l), j, 1-t)
), j=i..nops(l)))
end:
T:= n-> `if`(n=0, 1, (p-> seq(coeff(p, x, j), j=0..n))(
b([seq(irem(i, 2), i=2..n)], 1$2))):
seq(T(n), n=0..12);
MATHEMATICA
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[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]]];
Flatten[Table[T[n], {n, 0, 12}]] (* Jean-François Alcover, May 27 2018, from Maple *)
CROSSREFS
Columns k=0-10 give: A000007, A057427, A052955(n-2) for n>1, A305777, A305778, A305779, A305780, A305781, A305782, A305783, A305784.
Diagonals include A000012, A001477, A077043.
Row sums give A274547.
T(n,ceiling(n/2)) gives A305785.
Cf. A124419, A274310 (parities alternate within blocks), A305823.
Sequence in context: A277504 A167763 A277666 * A353279 A321919 A321918
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jun 29 2016
STATUS
approved