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A208475 Triangle read by rows: T(n,k) = total sum of odd/even parts >= k in all partitions of n, if k is odd/even.

%I A208475 #21 Mar 11 2015 07:27:17
%S A208475 1,2,2,7,2,3,10,10,3,4,23,12,11,4,5,36,30,17,14,5,6,65,40,35,18,17,6,
%T A208475 7,94,82,49,44,22,20,7,8,160,110,93,58,48,26,23,8,9,230,190,133,108,
%U A208475 70,56,30,26,9,10,356,260,217,148,124,76,64,34,29,10,11
%N A208475 Triangle read by rows: T(n,k) = total sum of odd/even parts >= k in all partitions of n, if k is odd/even.
%C A208475 Essentially this sequence is related to A206561 in the same way as A206563 is related to A181187. See the calculation in the example section of A206563.
%H A208475 Alois P. Heinz, <a href="/A208475/b208475.txt">Rows n = 1..141, flattened</a>
%e A208475 Triangle begins:
%e A208475 1;
%e A208475 2,   2;
%e A208475 7,   2,  3;
%e A208475 10, 10,  3,  4;
%e A208475 23, 12, 11,  4,  5;
%e A208475 36, 30, 17, 14,  5,  6;
%p A208475 p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
%p A208475 b:= proc(n, i) option remember; local f, g;
%p A208475       if n=0 then [1]
%p A208475     elif i=1 then [1, n]
%p A208475     else f:= b(n, i-1); g:= `if`(i>n, [0], b(n-i, i));
%p A208475          p (p (f, g), [0$i, g[1]])
%p A208475       fi
%p A208475     end:
%p A208475 T:= proc(n) local l;
%p A208475       l:= b(n, n);
%p A208475       seq (add (l[i+2*j+1]*(i+2*j), j=0..(n-i)/2), i=1..n)
%p A208475     end:
%p A208475 seq (T(n), n=1..14);  # _Alois P. Heinz_, Mar 21 2012
%t A208475 p[f_, g_] := With[{m = Max[Length[f], Length[g]]}, PadRight[f, m, 0] + PadRight[g, m, 0]]; b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1}, i == 1, {1, n}, True, f = b[n, i-1]; g = If[i>n, {0}, b[n-i, i]]; p[p[f, g], Append[Array[0&, i], g[[1]]]]]]; T[n_] := Module[{l}, l = b[n, n]; Table[Sum[l[[i+2j+1]]*(i+2j), {j, 0, (n-i)/2}], {i, 1, n}]]; Table[T[n], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Mar 11 2015, after _Alois P. Heinz_ *)
%Y A208475 Column 1-2: A066967, A066966. Right border is A000027.
%Y A208475 Cf. A138785, A181187, A206561, A206563, A208476.
%K A208475 nonn,tabl
%O A208475 1,2
%A A208475 _Omar E. Pol_, Feb 28 2012
%E A208475 More terms from _Alois P. Heinz_, Mar 21 2012