A208475 - OEIS

%I #21 Mar 11 2015 07:27:17

%S 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 7,94,82,49,44,22,20,7,8,160,110,93,58,48,26,23,8,9,230,190,133,108,

%U 70,56,30,26,9,10,356,260,217,148,124,76,64,34,29,10,11

%N 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 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 Alois P. Heinz, <a href="/A208475/b208475.txt">Rows n = 1..141, flattened</a>

%e Triangle begins:

%e 1;

%e 2,   2;

%e 7,   2,  3;

%e 10, 10,  3,  4;

%e 23, 12, 11,  4,  5;

%e 36, 30, 17, 14,  5,  6;

%p p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):

%p b:= proc(n, i) option remember; local f, g;

%p       if n=0 then [1]

%p     elif i=1 then [1, n]

%p     else f:= b(n, i-1); g:= `if`(i>n, [0], b(n-i, i));

%p          p (p (f, g), [0$i, g[1]])

%p       fi

%p     end:

%p T:= proc(n) local l;

%p       l:= b(n, n);

%p       seq (add (l[i+2*j+1]*(i+2*j), j=0..(n-i)/2), i=1..n)

%p     end:

%p seq (T(n), n=1..14);  # _Alois P. Heinz_, Mar 21 2012

%t 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 Column 1-2: A066967, A066966. Right border is A000027.

%Y Cf. A138785, A181187, A206561, A206563, A208476.

%K nonn,tabl

%O 1,2

%A _Omar E. Pol_, Feb 28 2012

%E More terms from _Alois P. Heinz_, Mar 21 2012