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A210947 Triangle read by rows: T(n,k) = total number of parts <= k of all partitions of n. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A266646 A102455 A073678 * A031981 A001476 A279486

Adjacent sequences:  A210944 A210945 A210946 * A210948 A210949 A210950

KEYWORD

nonn,tabl

AUTHOR

Omar E. Pol, May 01 2012

STATUS

approved

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Last modified November 20 14:54 EST 2019. Contains 329337 sequences. (Running on oeis4.)