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A210485 Number T(n,k) of parts in all partitions of n in which no part occurs more than k times; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 6
0, 0, 1, 0, 1, 3, 0, 3, 3, 6, 0, 3, 8, 8, 12, 0, 5, 11, 15, 15, 20, 0, 8, 17, 24, 29, 29, 35, 0, 10, 23, 36, 41, 47, 47, 54, 0, 13, 36, 50, 65, 71, 78, 78, 86, 0, 18, 48, 75, 91, 104, 111, 119, 119, 128, 0, 25, 69, 102, 132, 150, 165, 173, 182, 182, 192 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

T(n,k) is defined for n,k >= 0. The triangle contains terms with k <= n. T(n,k) = T(n,n) = A006128(n) for k >= n.

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

FORMULA

T(n,k) = Sum_{i=0..k} A213177(n,i).

EXAMPLE

T(6,2) = 17: [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [2,2,1,1].

Triangle T(n,k) begins:

0;

0,  1;

0,  1,  3;

0,  3,  3,  6;

0,  3,  8,  8, 12;

0,  5, 11, 15, 15, 20;

0,  8, 17, 24, 29, 29, 35;

0, 10, 23, 36, 41, 47, 47, 54;

0, 13, 36, 50, 65, 71, 78, 78, 86;

MAPLE

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],

      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))

    end:

T:= (n, k)-> b(n, n, k)[2]:

seq(seq(T(n, k), k=0..n), n=0..12);

MATHEMATICA

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[b[n-i*j, i-1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]]*j}, {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[n, n, k][[2]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Dec 27 2013, translated from Maple *)

CROSSREFS

Columns k=0-3 give: A000004, A015723, A185350, A117148.

Diagonal gives: A006128.

Cf. A213177.

Sequence in context: A078907 A282135 A278923 * A111815 A281269 A210877

Adjacent sequences:  A210482 A210483 A210484 * A210486 A210487 A210488

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jan 23 2013

STATUS

approved

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Last modified October 17 01:24 EDT 2018. Contains 316275 sequences. (Running on oeis4.)