A340991 Triangle T(n,k) whose k-th column is the k-fold self-convolution of the primes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 2, 0, 3, 4, 0, 5, 12, 8, 0, 7, 29, 36, 16, 0, 11, 58, 114, 96, 32, 0, 13, 111, 291, 376, 240, 64, 0, 17, 188, 669, 1160, 1120, 576, 128, 0, 19, 305, 1386, 3121, 4040, 3120, 1344, 256, 0, 23, 462, 2678, 7532, 12450, 12864, 8288, 3072, 512, 0, 29, 679, 4851, 16754, 34123, 44652, 38416, 21248, 6912, 1024

Offset: 0

Author

Alois P. Heinz, Feb 01 2021

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
  1;
  0,  2;
  0,  3,   4;
  0,  5,  12,    8;
  0,  7,  29,   36,   16;
  0, 11,  58,  114,   96,    32;
  0, 13, 111,  291,  376,   240,    64;
  0, 17, 188,  669, 1160,  1120,   576,  128;
  0, 19, 305, 1386, 3121,  4040,  3120, 1344,  256;
  0, 23, 462, 2678, 7532, 12450, 12864, 8288, 3072, 512;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Columns k=0-4 give (offsets may differ): A000007, A000040, A014342, A014343, A014344.

Main diagonal gives A000079.

Row sums give A030017(n+1).

T(2n,n) gives A340990.

Cf. A030018, A030281.

Programs

Maple

T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, ithprime(n)), (q->
       add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);
# Uses function PMatrix from A357368.
PMatrix(10, ithprime); # Peter Luschny, Oct 09 2022

Mathematica

T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
     If[k == 1, If[n == 0, 0, Prime[n]], With[{q = Quotient[k, 2]},
     Sum[T[j, q] T[n - j, k - q], {j, 0, n}]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 10 2021, after Alois P. Heinz *)

Formula

T(n,k) = [x^n] (Sum_{j>=1} prime(j)*x^j)^k.

Sum_{k=0..n} k * T(n,k) = A030281(n).

Sum_{k=0..n} (-1)^k * T(n,k) = A030018(n).

Conjecture: row polynomials are x*R(n,1) for n > 0 where R(n,k) = R(n-1,k+1) + x*R(n-1,1)*R(1,k) for n > 1, k > 0 with R(1,k) = prime(k) for k > 0. The same recursion seems to work for self-convolution of any other sequence. - Mikhail Kurkov, Apr 05 2025