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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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: A258278 A122856 A055791 * A191400 A168315 A120730

Adjacent sequences:  A245839 A245840 A245841 * A245843 A245844 A245845

KEYWORD

nonn,tabl

AUTHOR

L. Edson Jeffery, Aug 03 2014

STATUS

approved

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Last modified February 19 02:06 EST 2018. Contains 299330 sequences. (Running on oeis4.)