A245842 Triangle T read by rows: T(n,k) = Total number of even parts in all partitions of n with exactly k parts, 1 <= k <= n.

0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 0, 2, 2, 1, 0, 1, 2, 5, 2, 1, 0, 0, 3, 4, 5, 2, 1, 0, 1, 4, 7, 8, 5, 2, 1, 0, 0, 4, 8, 10, 8, 5, 2, 1, 0, 1, 4, 12, 14, 15, 8, 5, 2, 1, 0, 0, 5, 12, 19, 18, 15, 8, 5, 2, 1, 0, 1, 6, 18, 24, 27, 24, 15, 8, 5, 2, 1, 0

Offset

1,8

Comments

Column sequences appear to converge to A066897.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

T(n,k) + A245840(n,k) = A172467(n,k).

Example

Triangle begins
0
1  0
0  1   0
1  2   1   0
0  2   2   1   0
1  2   5   2   1   0
0  3   4   5   2   1  0
1  4   7   8   5   2  1  0
0  4   8  10   8   5  2  1  0
1  4  12  14  15   8  5  2  1  0
0  5  12  19  18  15  8  5  2  1  0

Maple

b:= proc(n, i, k) option remember; `if`(n=0, [`if`(k=0, 1, 0), 0],
      `if`(i<1 or k=0, [0$2], ((f, g)-> f+g+[0, `if`(irem(i, 2)=0,
       g[1], 0)])(b(n, i-1, k), `if`(i>n, [0$2], b(n-i, i, k-1)))))
    end:
T:= (n, k)-> b(n$2, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Aug 04 2014

Mathematica

Grid[Table[Sum[Count[Flatten[IntegerPartitions[n, {k}]], i], {i, 2, n, 2}], {n, 11}, {k, n}]]
(* second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, {If[k == 0, 1, 0], 0}, If[i < 1 || k == 0, {0, 0}, Function[{f, g}, f + g + {0, If[Mod[i, 2] == 0, g[[1]], 0]}][b[n, i-1, k], If[i > n, {0, 0}, b[n-i, i, k-1]]]]];
T[n_, k_] := b[n, n, k][[2]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, May 21 2016, after Alois P. Heinz *)

Crossrefs

Cf. A245843 (partial sums of row entries).

Cf. A066898 (row sums), A172467.

Cf. A245840, A245841.

Sequence in context: A328795 A353846 A055791 * A300574 A352194 A327757

Adjacent sequences: A245839 A245840 A245841 * A245843 A245844 A245845

Keyword

nonn,tabl

Author

L. Edson Jeffery, Aug 03 2014

Status

approved