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

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, 92, 98, 105, 45, 83, 110, 134, 149, 161, 168, 176, 67, 119, 164, 196, 221, 239, 253, 261, 270, 97, 179, 242, 294, 334, 364, 385, 401, 410, 420

Offset: 1

Author

Omar E. Pol, May 01 2012

Keywords

• nonn
• tabl

Comments

Row n lists the partial sums of row n of triangle A138785.

Examples

			Triangle begins:
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,  92,  98, 105;
45,  83, 110, 134, 149, 161, 168, 176;
67, 119, 164, 196, 221, 239, 253, 261, 270;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Column 1 is A000070(n-1). Right border gives A066186.

Cf. A181187, A138785, A206561, A210948, A208475, A210956.

Programs

Maple

p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
b:= proc(n, i) option remember; local f, g;
      if n=0 then [1]
    elif i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0], b(n-i, i));
         p (p (f, g), [0$i, g[1]*i])
      fi
    end:
T:= proc(n, k) option remember;
       b(n, n)[k+1] +`if`(k<2, 0, T(n, k-1))
    end:
seq (seq (T(n,k), k=1..n), n=1..12);  # Alois P. Heinz, May 02 2012

Mathematica

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 *)

Formula

T(n,k) = sum_{j=1..k} A138785(n,j).