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A210948 Triangle read by rows: T(n,k) = sum of all parts <= k of all partitions of n.

%I A210948 #18 Mar 11 2015 07:27:08
%S A210948 1,2,4,4,6,9,7,13,16,20,12,20,26,30,35,19,35,47,55,60,66,30,52,70,82,
%T A210948 92,98,105,45,83,110,134,149,161,168,176,67,119,164,196,221,239,253,
%U A210948 261,270,97,179,242,294,334,364,385,401,410,420
%N A210948 Triangle read by rows: T(n,k) = sum of all parts <= k of all partitions of n.
%C A210948 Row n lists the partial sums of row n of triangle A138785.
%H A210948 Alois P. Heinz, <a href="/A210948/b210948.txt">Rows n = 1..141, flattened</a>
%F A210948 T(n,k) = sum_{j=1..k} A138785(n,j).
%e A210948 Triangle begins:
%e A210948 1;
%e A210948 2,    4;
%e A210948 4,    6,   9;
%e A210948 7,   13,  16,  20;
%e A210948 12,  20,  26,  30,  35;
%e A210948 19,  35,  47,  55,  60,  66;
%e A210948 30,  52,  70,  82,  92,  98, 105;
%e A210948 45,  83, 110, 134, 149, 161, 168, 176;
%e A210948 67, 119, 164, 196, 221, 239, 253, 261, 270;
%p A210948 p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
%p A210948 b:= proc(n, i) option remember; local f, g;
%p A210948       if n=0 then [1]
%p A210948     elif i=1 then [1, n]
%p A210948     else f:= b(n, i-1); g:= `if`(i>n, [0], b(n-i, i));
%p A210948          p (p (f, g), [0$i, g[1]*i])
%p A210948       fi
%p A210948     end:
%p A210948 T:= proc(n, k) option remember;
%p A210948        b(n, n)[k+1] +`if`(k<2, 0, T(n, k-1))
%p A210948     end:
%p A210948 seq (seq (T(n,k), k=1..n), n=1..12);  # _Alois P. Heinz_, May 02 2012
%t A210948 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], i*g[[1]]]]]]; T[n_, k_] := T[n, k] = b[n, n][[k+1]] + If[k<2, 0, T[n, k-1]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12 }] // Flatten (* _Jean-François Alcover_, Mar 11 2015, after _Alois P. Heinz_ *)
%Y A210948 Column 1 is A000070(n-1). Right border gives A066186.
%Y A210948 Cf. A181187, A138785, A206561, A210948, A208475, A210956.
%K A210948 nonn,tabl
%O A210948 1,2
%A A210948 _Omar E. Pol_, May 01 2012