A245841 Triangle T read by rows: T(n,k) = Total number of odd parts in all partitions of n with at most k parts, 1 <= k <= n.

1, 0, 2, 1, 2, 5, 0, 2, 4, 8, 1, 3, 7, 10, 15, 0, 4, 8, 14, 18, 24, 1, 4, 12, 19, 27, 32, 39, 0, 4, 12, 24, 34, 44, 50, 58, 1, 5, 18, 32, 49, 62, 74, 81, 90, 0, 6, 18, 40, 60, 82, 98, 112, 120, 130, 1, 6, 24, 49, 81, 108, 135, 154, 170, 179, 190

Offset

1,3

Links

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

Example

Triangle begins:
1
0  2
1  2   5
0  2   4   8
1  3   7  10  15
0  4   8  14  18   24
1  4  12  19  27   32   39
0  4  12  24  34   44   50   58
1  5  18  32  49   62   74   81   90
0  6  18  40  60   82   98  112  120  130
1  6  24  49  81  108  135  154  170  179  190

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:= 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, 1, 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] == 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}] // Accumulate, {n, 1, 14}] // Flatten (* Jean-François Alcover, Jul 21 2016, after Alois P. Heinz *)

Crossrefs

Partial sums of row entries of A245840.

Cf. A066897 (outer diagonal).

Cf. A245842, A245843.

Sequence in context: A032259 A109851 A371136 * A011404 A002211 A308947

Adjacent sequences: A245838 A245839 A245840 * A245842 A245843 A245844

Keyword

nonn,tabl

Author

L. Edson Jeffery, Aug 03 2014

Status

approved