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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.
4

%I #12 May 21 2016 03:04:24

%S 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,

%T 12,20,25,27,28,28,0,4,12,22,30,35,37,38,38,1,5,17,31,46,54,59,61,62,

%U 62,0,5,17,36,54,69,77,82,84,85,85

%N 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.

%H Alois P. Heinz, <a href="/A245843/b245843.txt">Rows n = 1..141, flattened</a>

%e Triangle begins:

%e 0

%e 1 1

%e 0 1 1

%e 1 3 4 4

%e 0 2 4 5 5

%e 1 3 8 10 11 11

%e 0 3 7 12 14 15 15

%e 1 5 12 20 25 27 28 28

%e 0 4 12 22 30 35 37 38 38

%e 1 5 17 31 46 54 59 61 62 62

%e 0 5 17 36 54 69 77 82 84 85 85

%p b:= proc(n, i, k) option remember; `if`(n=0, [`if`(k=0, 1, 0), 0],

%p `if`(i<1 or k=0, [0$2], ((f, g)-> f+g+[0, `if`(irem(i, 2)=0,

%p g[1], 0)])(b(n, i-1, k), `if`(i>n, [0$2], b(n-i, i, k-1)))))

%p end:

%p T:= proc(n, k) T(n, k):= b(n$2, k)[2]+`if`(k=1, 0, T(n, k-1)) end:

%p seq(seq(T(n, k), k=1..n), n=1..14); # _Alois P. Heinz_, Aug 04 2014

%t Grid[Table[Sum[Sum[Count[Flatten[IntegerPartitions[n, {j}]], i], {i, 2, n, 2}], {j, k}], {n, 11}, {k, n}]]

%t (* second program: *)

%t 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 T[n_, k_] := b[n, n, k][[2]] + If[k == 1, 0, T[n, k-1]];

%t Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, May 21 2016, after _Alois P. Heinz_ *)

%Y Partial sums of row entries of A245842.

%Y Cf. A066898 (outer diagonal).

%Y Cf. A245840, A245841.

%K nonn,tabl

%O 1,8

%A _L. Edson Jeffery_, Aug 03 2014