A340991 - OEIS

%I #21 Oct 09 2022 05:35:18

%S 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,

%T 240,64,0,17,188,669,1160,1120,576,128,0,19,305,1386,3121,4040,3120,

%U 1344,256,0,23,462,2678,7532,12450,12864,8288,3072,512,0,29,679,4851,16754,34123,44652,38416,21248,6912,1024

%N 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.

%H Alois P. Heinz, <a href="/A340991/b340991.txt">Rows n = 0..200, flattened</a>

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

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

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

%e Triangle T(n,k) begins:

%e   1;

%e   0,  2;

%e   0,  3,   4;

%e   0,  5,  12,    8;

%e   0,  7,  29,   36,   16;

%e   0, 11,  58,  114,   96,    32;

%e   0, 13, 111,  291,  376,   240,    64;

%e   0, 17, 188,  669, 1160,  1120,   576,  128;

%e   0, 19, 305, 1386, 3121,  4040,  3120, 1344,  256;

%e   0, 23, 462, 2678, 7532, 12450, 12864, 8288, 3072, 512;

%e   ...

%p T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),

%p       `if`(k=1, `if`(n=0, 0, ithprime(n)), (q->

%p        add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))

%p     end:

%p seq(seq(T(n, k), k=0..n), n=0..12);

%p # Uses function PMatrix from A357368.

%p PMatrix(10, ithprime); # _Peter Luschny_, Oct 09 2022

%t T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],

%t      If[k == 1, If[n == 0, 0, Prime[n]], With[{q = Quotient[k, 2]},

%t      Sum[T[j, q] T[n - j, k - q], {j, 0, n}]]]];

%t Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 10 2021, after _Alois P. Heinz_ *)

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

%Y Main diagonal gives A000079.

%Y Row sums give A030017(n+1).

%Y T(2n,n) gives A340990.

%Y Cf. A030018, A030281.

%K nonn,tabl

%O 0,3

%A _Alois P. Heinz_, Feb 01 2021