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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.

%I A245841 #14 Jul 21 2016 06:13:44
%S A245841 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,
%T A245841 4,12,24,34,44,50,58,1,5,18,32,49,62,74,81,90,0,6,18,40,60,82,98,112,
%U A245841 120,130,1,6,24,49,81,108,135,154,170,179,190
%N 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.
%H A245841 Alois P. Heinz, <a href="/A245841/b245841.txt">Rows n = 1..141, flattened</a>
%e A245841 Triangle begins:
%e A245841 1
%e A245841 0  2
%e A245841 1  2   5
%e A245841 0  2   4   8
%e A245841 1  3   7  10  15
%e A245841 0  4   8  14  18   24
%e A245841 1  4  12  19  27   32   39
%e A245841 0  4  12  24  34   44   50   58
%e A245841 1  5  18  32  49   62   74   81   90
%e A245841 0  6  18  40  60   82   98  112  120  130
%e A245841 1  6  24  49  81  108  135  154  170  179  190
%p A245841 b:= proc(n, i, k) option remember; `if`(n=0, [`if`(k=0, 1, 0), 0],
%p A245841       `if`(i<1 or k=0, [0$2], ((f, g)-> f+g+[0, `if`(irem(i, 2)=1,
%p A245841        g[1], 0)])(b(n, i-1, k), `if`(i>n, [0$2], b(n-i, i, k-1)))))
%p A245841     end:
%p A245841 T:= proc(n, k) T(n, k):= b(n$2, k)[2]+`if`(k=1, 0, T(n, k-1)) end:
%p A245841 seq(seq(T(n, k), k=1..n), n=1..14);  # _Alois P. Heinz_, Aug 04 2014
%t A245841 Grid[Table[Sum[Sum[Count[Flatten[IntegerPartitions[n, {j}]], i], {i, 1, n, 2}], {j, k}], {n, 11}, {k, n}]]
%t A245841 (* second program: *)
%t A245841 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 A245841 T[n_, k_] := b[n, n, k][[2]];
%t A245841 Table[Table[T[n, k], {k, 1, n}] // Accumulate, {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Jul 21 2016, after _Alois P. Heinz_ *)
%Y A245841 Partial sums of row entries of A245840.
%Y A245841 Cf. A066897 (outer diagonal).
%Y A245841 Cf. A245842, A245843.
%K A245841 nonn,tabl
%O A245841 1,3
%A A245841 _L. Edson Jeffery_, Aug 03 2014