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URL: https://oeis.org/A363048

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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.
5
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, 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, 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
OFFSET
0,5
LINKS
FORMULA
For k > 0, g.f. of column k: Sum_{i>=0} x^(k*i)/Product_{j=1..k*i} (1-x^j).
EXAMPLE
Triangle begins:
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, 2, 1, 1;
0, 22, 12, 7, 6, 3, 2, 1, 1;
0, 30, 14, 11, 7, 5, 3, 2, 1, 1;
...
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
end:
T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), add(
(j-> b(n-j, min(n-j, j)))(k*i), i=0..n/k)):
seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, May 14 2023
MATHEMATICA
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[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}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Oct 20 2023, after Alois P. Heinz *)
PROG
(PARI) T(n, k) = sum(j=0, n, #partitions(n-k*j, k*j));
CROSSREFS
Row sums give A323433.
Column k=0..5 give A000007, A000041, A027187, A363045, A363046, A363047.
T(2n,n) gives A052810.
Sequence in context: A381886 A167192 A335435 * A262114 A320780 A391984
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, May 14 2023
STATUS
approved