A245840 Triangle T read by rows: T(n,k) = Total number of odd parts in all partitions of n with exactly k parts, 1 <= k <= n.

1, 0, 2, 1, 1, 3, 0, 2, 2, 4, 1, 2, 4, 3, 5, 0, 4, 4, 6, 4, 6, 1, 3, 8, 7, 8, 5, 7, 0, 4, 8, 12, 10, 10, 6, 8, 1, 4, 13, 14, 17, 13, 12, 7, 9, 0, 6, 12, 22, 20, 22, 16, 14, 8, 10, 1, 5, 18, 25, 32, 27, 27, 19, 16, 9, 11

Offset: 1

Author

L. Edson Jeffery, Aug 03 2014

Keywords

• nonn
• tabl

Examples

			Triangle begins
1
0  2
1  1   3
0  2   2   4
1  2   4   3   5
0  4   4   6   4   6
1  3   8   7   8   5   7
0  4   8  12  10  10   6   8
1  4  13  14  17  13  12   7   9
0  6  12  22  20  22  16  14   8 10
1  5  18  25  32  27  27  19  16  9  11
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Cf. A066897 (row sums), A245841 (partial sums of row entries).

Cf. A245842, A245843.

Programs

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)=1,
       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, 1, n, 2}], {n, 1, 11}, {k, 1, n}]]
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]==1, 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, Nov 17 2015, after Alois P. Heinz *)

Formula

T(n,k) + A245842(n,k) = A172467(n,k).