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A245840 Triangle T read by rows: T(n,k) = Total number of odd parts in all partitions of n with exactly k parts, 1 <= k <= n.

%I A245840 #16 Nov 17 2015 04:11:10
%S A245840 1,0,2,1,1,3,0,2,2,4,1,2,4,3,5,0,4,4,6,4,6,1,3,8,7,8,5,7,0,4,8,12,10,
%T A245840 10,6,8,1,4,13,14,17,13,12,7,9,0,6,12,22,20,22,16,14,8,10,1,5,18,25,
%U A245840 32,27,27,19,16,9,11
%N A245840 Triangle T read by rows: T(n,k) = Total number of odd parts in all partitions of n with exactly k parts, 1 <= k <= n.
%H A245840 Alois P. Heinz, <a href="/A245840/b245840.txt">Rows n = 1..141, flattened</a>
%F A245840 T(n,k) + A245842(n,k) = A172467(n,k).
%e A245840 Triangle begins
%e A245840 1
%e A245840 0  2
%e A245840 1  1   3
%e A245840 0  2   2   4
%e A245840 1  2   4   3   5
%e A245840 0  4   4   6   4   6
%e A245840 1  3   8   7   8   5   7
%e A245840 0  4   8  12  10  10   6   8
%e A245840 1  4  13  14  17  13  12   7   9
%e A245840 0  6  12  22  20  22  16  14   8 10
%e A245840 1  5  18  25  32  27  27  19  16  9  11
%p A245840 b:= proc(n, i, k) option remember; `if`(n=0, [`if`(k=0, 1, 0), 0],
%p A245840       `if`(i<1 or k=0, [0$2], ((f, g)-> f+g+[0, `if`(irem(i, 2)=1,
%p A245840        g[1], 0)])(b(n, i-1, k), `if`(i>n, [0$2], b(n-i, i, k-1)))))
%p A245840     end:
%p A245840 T:= (n, k)-> b(n$2, k)[2]:
%p A245840 seq(seq(T(n, k), k=1..n), n=1..14);  # _Alois P. Heinz_, Aug 04 2014
%t A245840 Grid[Table[Sum[Count[Flatten[IntegerPartitions[n, {k}]], i], {i, 1, n, 2}], {n, 1, 11}, {k, 1, n}]]
%t A245840 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[n_, k_] := b[n, n, k][[2]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Nov 17 2015, after _Alois P. Heinz_ *)
%Y A245840 Cf. A066897 (row sums), A245841 (partial sums of row entries).
%Y A245840 Cf. A245842, A245843.
%K A245840 nonn,tabl
%O A245840 1,3
%A A245840 _L. Edson Jeffery_, Aug 03 2014