A210947 Triangle read by rows: T(n,k) = total number of parts <= k of all partitions of n.

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 12, 16, 18, 19, 20, 19, 27, 31, 33, 34, 35, 30, 41, 47, 50, 52, 53, 54, 45, 64, 73, 79, 82, 84, 85, 86, 67, 93, 108, 116, 121, 124, 126, 127, 128, 97, 138, 159, 172, 180, 185, 188, 190, 191, 192

Offset

1,2

Comments

Row n lists the partial sums of row n of triangle A066633.

Links

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

Formula

T(n,k) = Sum_{j=1..k} A066633(n,j).

Example

Triangle begins:
1;
2,   3;
4,   5,  6;
7,  10,  11,  12;
12, 16,  18,  19,  20;
19, 27,  31,  33,  34,  35;
30, 41,  47,  50,  52,  53,  54;
45, 64,  73,  79,  82,  84,  85,  86;
67, 93, 108, 116, 121, 124, 126, 127, 128;

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]])
      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..11); # 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}, If[n == 0, {1}, If[i == 1, {1, n}, 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_, 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, 11}] // Flatten (* Jean-François Alcover, Mar 11 2015, after Alois P. Heinz *)

Crossrefs

Column 1 is A000070(n-1). Right border gives A006128.

Cf. A066633, A181187, A206563, A210948, A210955.

Sequence in context: A362854 A102455 A073678 * A031981 A279486 A001476

Adjacent sequences: A210944 A210945 A210946 * A210948 A210949 A210950

Keyword

nonn,tabl

Author

Omar E. Pol, May 01 2012

Status

approved