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

0,3

Links

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

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).

Example

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;
  ...

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 *)

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.

Sequence in context: A117909 A261094 A091538 * A013584 A307320 A350227

Adjacent sequences: A340988 A340989 A340990 * A340992 A340993 A340994

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 01 2021

Status

approved