A245843 Triangle T read by rows: T(n,k) = Total number of even parts in all partitions of n with at most k parts, 1 <= k <= n.

0, 1, 1, 0, 1, 1, 1, 3, 4, 4, 0, 2, 4, 5, 5, 1, 3, 8, 10, 11, 11, 0, 3, 7, 12, 14, 15, 15, 1, 5, 12, 20, 25, 27, 28, 28, 0, 4, 12, 22, 30, 35, 37, 38, 38, 1, 5, 17, 31, 46, 54, 59, 61, 62, 62, 0, 5, 17, 36, 54, 69, 77, 82, 84, 85, 85

Offset: 1

Author

L. Edson Jeffery, Aug 03 2014

Keywords

• nonn
• tabl

Examples

			Triangle begins:
0
1  1
0  1   1
1  3   4   4
0  2   4   5   5
1  3   8  10  11  11
0  3   7  12  14  15  15
1  5  12  20  25  27  28  28
0  4  12  22  30  35  37  38  38
1  5  17  31  46  54  59  61  62  62
0  5  17  36  54  69  77  82  84  85  85
		

Links

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

Crossrefs

Partial sums of row entries of A245842.

Cf. A066898 (outer diagonal).

Cf. A245840, A245841.

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

Mathematica

Grid[Table[Sum[Sum[Count[Flatten[IntegerPartitions[n, {j}]], i], {i, 2, n, 2}], {j, k}], {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]] + If[k == 1, 0, T[n, k-1]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, May 21 2016, after Alois P. Heinz *)