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

1,2

Comments

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

Links

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

Formula

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

Example

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;

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 *)

Crossrefs

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

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

Sequence in context: A325245 A241064 A292671 * A359678 A008133 A237828

Adjacent sequences: A210945 A210946 A210947 * A210949 A210950 A210951

Keyword

nonn,tabl

Author

Omar E. Pol, May 01 2012

Status

approved