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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.

%I A245843 #12 May 21 2016 03:04:24
%S A245843 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 A245843 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 A245843 62,0,5,17,36,54,69,77,82,84,85,85
%N 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.
%H A245843 Alois P. Heinz, <a href="/A245843/b245843.txt">Rows n = 1..141, flattened</a>
%e A245843 Triangle begins:
%e A245843 0
%e A245843 1  1
%e A245843 0  1   1
%e A245843 1  3   4   4
%e A245843 0  2   4   5   5
%e A245843 1  3   8  10  11  11
%e A245843 0  3   7  12  14  15  15
%e A245843 1  5  12  20  25  27  28  28
%e A245843 0  4  12  22  30  35  37  38  38
%e A245843 1  5  17  31  46  54  59  61  62  62
%e A245843 0  5  17  36  54  69  77  82  84  85  85
%p A245843 b:= proc(n, i, k) option remember; `if`(n=0, [`if`(k=0, 1, 0), 0],
%p A245843       `if`(i<1 or k=0, [0$2], ((f, g)-> f+g+[0, `if`(irem(i, 2)=0,
%p A245843        g[1], 0)])(b(n, i-1, k), `if`(i>n, [0$2], b(n-i, i, k-1)))))
%p A245843     end:
%p A245843 T:= proc(n, k) T(n, k):= b(n$2, k)[2]+`if`(k=1, 0, T(n, k-1)) end:
%p A245843 seq(seq(T(n, k), k=1..n), n=1..14);  # _Alois P. Heinz_, Aug 04 2014
%t A245843 Grid[Table[Sum[Sum[Count[Flatten[IntegerPartitions[n, {j}]], i], {i, 2, n, 2}], {j, k}], {n, 11}, {k, n}]]
%t A245843 (* second program: *)
%t A245843 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 A245843 T[n_, k_] := b[n, n, k][[2]] + If[k == 1, 0, T[n, k-1]];
%t A245843 Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, May 21 2016, after _Alois P. Heinz_ *)
%Y A245843 Partial sums of row entries of A245842.
%Y A245843 Cf. A066898 (outer diagonal).
%Y A245843 Cf. A245840, A245841.
%K A245843 nonn,tabl
%O A245843 1,8
%A A245843 _L. Edson Jeffery_, Aug 03 2014