%I #25 Jul 15 2023 17:33:36

%S 0,0,1,0,1,2,0,3,0,3,0,3,5,0,4,0,5,6,4,0,5,0,8,9,7,5,0,6,0,10,13,13,5,

%T 6,0,7,0,13,23,14,15,6,7,0,8,0,18,30,27,16,13,7,8,0,9,0,25,44,33,30,

%U 18,15,8,9,0,10,0,30,58,55,36,34,15,17,9,10,0,11

%N Number T(n,k) of parts in all partitions of n with largest multiplicity k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%H Alois P. Heinz, <a href="/A213177/b213177.txt">Rows n = 0..140, flattened</a>

%F T(n,k) = A210485(n,k) - A210485(n,k-1) for k>0, T(n,0) = 0.

%e T(6,1) = 8: partitions of 6 with largest multiplicity 1 are [3,2,1], [4,2], [5,1], [6], with 3+2+2+1 = 8 parts.

%e T(6,2) = 9: [2,2,1,1], [3,3], [4,1,1].

%e T(6,3) = 7: [2,2,2], [3,1,1,1].

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

%e T(6,5) = 0.

%e T(6,6) = 6: [1,1,1,1,1,1].

%e Triangle begins:

%e 0;

%e 0, 1;

%e 0, 1, 2;

%e 0, 3, 0, 3;

%e 0, 3, 5, 0, 4;

%e 0, 5, 6, 4, 0, 5;

%e 0, 8, 9, 7, 5, 0, 6;

%e 0, 10, 13, 13, 5, 6, 0, 7;

%e 0, 13, 23, 14, 15, 6, 7, 0, 8;

%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] -b(n, n, k-1)[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[_, 0] = 0; T[n_, k_] := b[n, n, k][[2]] - b[n, n, k-1][[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, A320372, A320373, A320374, A320375, A320376, A320377, A320378, A320379, A320380.

%Y Row sums give: A006128.

%Y Main diagonal and first lower diagonal give: A001477, A063524.

%Y T(2n,n) gives A320381.

%Y Cf. A091602, A210485.

%K nonn,tabl

%O 0,6

%A _Alois P. Heinz_, Feb 27 2013