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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. 14
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.

For fixed k > 0, T(n,k) ~ 3^(1/4) * log(k+1) * exp(Pi*sqrt(2*k*n/(3*(k+1)))) / (Pi * (8*k*(k+1)*n)^(1/4)). - Vaclav Kotesovec, Oct 18 2018

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-François Alcover, Dec 27 2013, translated from Maple *)

CROSSREFS

Columns k=0-10 give: A000004, A015723, A185350, A117148, A320607, A320608, A320609, A320610, A320611, A320612, A320613.

Main diagonal gives: A006128.

Cf. A213177, A286653.

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 January 23 17:13 EST 2019. Contains 319399 sequences. (Running on oeis4.)