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A363048 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) is the number of partitions of n whose greatest part is a multiple of k.

%I A363048 #42 Oct 20 2023 06:45:15
%S A363048 1,0,1,0,2,1,0,3,1,1,0,5,3,1,1,0,7,3,2,1,1,0,11,6,4,2,1,1,0,15,7,5,3,
%T A363048 2,1,1,0,22,12,7,6,3,2,1,1,0,30,14,11,7,5,3,2,1,1,0,42,22,14,11,8,5,3,
%U A363048 2,1,1,0,56,27,19,14,11,7,5,3,2,1,1,0,77,40,27,21,15,12,7,5,3,2,1,1
%N A363048 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) is the number of partitions of n whose greatest part is a multiple of k.
%H A363048 Alois P. Heinz, <a href="/A363048/b363048.txt">Rows n = 0..200, flattened</a>
%F A363048 For k > 0, g.f. of column k: Sum_{i>=0} x^(k*i)/Product_{j=1..k*i} (1-x^j).
%e A363048 Triangle begins:
%e A363048   1;
%e A363048   0,  1;
%e A363048   0,  2,  1;
%e A363048   0,  3,  1,  1;
%e A363048   0,  5,  3,  1, 1;
%e A363048   0,  7,  3,  2, 1, 1;
%e A363048   0, 11,  6,  4, 2, 1, 1;
%e A363048   0, 15,  7,  5, 3, 2, 1, 1;
%e A363048   0, 22, 12,  7, 6, 3, 2, 1, 1;
%e A363048   0, 30, 14, 11, 7, 5, 3, 2, 1, 1;
%e A363048   ...
%p A363048 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A363048      `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
%p A363048     end:
%p A363048 T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), add(
%p A363048     (j-> b(n-j, min(n-j, j)))(k*i), i=0..n/k)):
%p A363048 seq(seq(T(n, k), k=0..n), n=0..12);  # _Alois P. Heinz_, May 14 2023
%t A363048 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]];
%t A363048 T[n_, k_] := If[k == 0, If[n == 0, 1, 0], Sum[Function[j, b[n - j, Min[n - j, j]]][k*i], {i, 0, n/k}]];
%t A363048 Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Oct 20 2023, after _Alois P. Heinz_ *)
%o A363048 (PARI) T(n, k) = sum(j=0, n, #partitions(n-k*j, k*j));
%Y A363048 Row sums give A323433.
%Y A363048 Column k=0..5 give A000007, A000041, A027187, A363045, A363046, A363047.
%Y A363048 T(2n,n) gives A052810.
%Y A363048 Cf. A008284, A072233, A350890.
%K A363048 nonn,tabl
%O A363048 0,5
%A A363048 _Seiichi Manyama_, May 14 2023