A299730 Irregular triangle read by rows: T(n,k) is the number of partitions of 3*n having exactly k prime parts; n >= 0, 0 <= k <= floor( 3*n / 2 ).

1, 1, 2, 3, 4, 3, 1, 6, 9, 8, 5, 2, 12, 20, 19, 14, 8, 3, 1, 19, 41, 42, 34, 21, 12, 5, 2, 37, 72, 88, 74, 53, 31, 18, 8, 3, 1, 58, 136, 161, 155, 115, 77, 46, 25, 12, 5, 2, 102, 226, 307, 291, 241, 168, 110, 65, 35, 18, 8, 3, 1

Offset: 0

Author

J. Stauduhar, Feb 17 2018

Keywords

• nonn
• tabf

Comments

Sequence of row lengths = A001651.

Examples

			The irregular triangle T(n, k) begins:
3n\k  0   1   2   3   4   5   6   7   8   9
0:    1
3:    1   2
6:    3   4   3   1
9:    6   9   8   5   2
12:  12  20  19  14   8   3   1
15:  19  41  42  34  21  12   5   2
18:  37  72  88  74  53  31  18   8   3   1
		

Links

• J. Stauduhar, Table of n, a(n) for n = 0..719

Crossrefs

Cf. A001651, A008585, A222656.

Programs

Maple

b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, 1,
      add(b(n-i*j, i-1)*`if`(isprime(i), x^j, 1), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(3*n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 03 2018

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, 1,
     Sum[b[n - i*j, i - 1]*If[PrimeQ[i], x^j, 1], {j, 0, n/i}]]];
T[n_] := CoefficientList[b[3n, 3n], x];
T /@ Range[0, 12] // Flatten (* Jean-François Alcover, Mar 08 2021, after Alois P. Heinz *)

Formula

T(n,k) = A222656(3n,k).