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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.

%I A245842 #22 May 21 2016 03:04:27
%S A245842 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,
%T A245842 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,
%U A245842 1,0,1,6,18,24,27,24,15,8,5,2,1,0
%N 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.
%C A245842 Column sequences appear to converge to A066897.
%H A245842 Alois P. Heinz, <a href="/A245842/b245842.txt">Rows n = 0..140, flattened</a>
%F A245842 T(n,k) + A245840(n,k) = A172467(n,k).
%e A245842 Triangle begins
%e A245842 0
%e A245842 1  0
%e A245842 0  1   0
%e A245842 1  2   1   0
%e A245842 0  2   2   1   0
%e A245842 1  2   5   2   1   0
%e A245842 0  3   4   5   2   1  0
%e A245842 1  4   7   8   5   2  1  0
%e A245842 0  4   8  10   8   5  2  1  0
%e A245842 1  4  12  14  15   8  5  2  1  0
%e A245842 0  5  12  19  18  15  8  5  2  1  0
%p A245842 b:= proc(n, i, k) option remember; `if`(n=0, [`if`(k=0, 1, 0), 0],
%p A245842       `if`(i<1 or k=0, [0$2], ((f, g)-> f+g+[0, `if`(irem(i, 2)=0,
%p A245842        g[1], 0)])(b(n, i-1, k), `if`(i>n, [0$2], b(n-i, i, k-1)))))
%p A245842     end:
%p A245842 T:= (n, k)-> b(n$2, k)[2]:
%p A245842 seq(seq(T(n, k), k=1..n), n=1..14);  # _Alois P. Heinz_, Aug 04 2014
%t A245842 Grid[Table[Sum[Count[Flatten[IntegerPartitions[n, {k}]], i], {i, 2, n, 2}], {n, 11}, {k, n}]]
%t A245842 (* second program: *)
%t A245842 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 A245842 T[n_, k_] := b[n, n, k][[2]];
%t A245842 Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, May 21 2016, after _Alois P. Heinz_ *)
%Y A245842 Cf. A245843 (partial sums of row entries).
%Y A245842 Cf. A066898 (row sums), A172467.
%Y A245842 Cf. A245840, A245841.
%K A245842 nonn,tabl
%O A245842 1,8
%A A245842 _L. Edson Jeffery_, Aug 03 2014