%I #30 Jul 16 2023 15:36:39

%S 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,

%T 10,23,36,41,47,47,54,0,13,36,50,65,71,78,78,86,0,18,48,75,91,104,111,

%U 119,119,128,0,25,69,102,132,150,165,173,182,182,192

%N 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.

%C 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.

%C 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

%H Alois P. Heinz, <a href="/A210485/b210485.txt">Rows n = 0..140, flattened</a>

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

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

%e Triangle T(n,k) begins:

%e 0;

%e 0, 1;

%e 0, 1, 3;

%e 0, 3, 3, 6;

%e 0, 3, 8, 8, 12;

%e 0, 5, 11, 15, 15, 20;

%e 0, 8, 17, 24, 29, 29, 35;

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

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

%e ...

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

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

%p end:

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

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

%t 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 *)

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

%Y Main diagonal gives A006128.

%Y T(2n,n) gives A364245.

%Y Cf. A213177, A286653.

%K nonn,tabl

%O 0,6

%A _Alois P. Heinz_, Jan 23 2013