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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.

%I A340991 #24 Apr 06 2025 14:53:28
%S A340991 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 A340991 240,64,0,17,188,669,1160,1120,576,128,0,19,305,1386,3121,4040,3120,
%U A340991 1344,256,0,23,462,2678,7532,12450,12864,8288,3072,512,0,29,679,4851,16754,34123,44652,38416,21248,6912,1024
%N 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.
%H A340991 Alois P. Heinz, <a href="/A340991/b340991.txt">Rows n = 0..200, flattened</a>
%F A340991 T(n,k) = [x^n] (Sum_{j>=1} prime(j)*x^j)^k.
%F A340991 Sum_{k=0..n} k * T(n,k) = A030281(n).
%F A340991 Sum_{k=0..n} (-1)^k * T(n,k) = A030018(n).
%F A340991 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
%e A340991 Triangle T(n,k) begins:
%e A340991   1;
%e A340991   0,  2;
%e A340991   0,  3,   4;
%e A340991   0,  5,  12,    8;
%e A340991   0,  7,  29,   36,   16;
%e A340991   0, 11,  58,  114,   96,    32;
%e A340991   0, 13, 111,  291,  376,   240,    64;
%e A340991   0, 17, 188,  669, 1160,  1120,   576,  128;
%e A340991   0, 19, 305, 1386, 3121,  4040,  3120, 1344,  256;
%e A340991   0, 23, 462, 2678, 7532, 12450, 12864, 8288, 3072, 512;
%e A340991   ...
%p A340991 T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
%p A340991       `if`(k=1, `if`(n=0, 0, ithprime(n)), (q->
%p A340991        add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A340991     end:
%p A340991 seq(seq(T(n, k), k=0..n), n=0..12);
%p A340991 # Uses function PMatrix from A357368.
%p A340991 PMatrix(10, ithprime); # _Peter Luschny_, Oct 09 2022
%t A340991 T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
%t A340991      If[k == 1, If[n == 0, 0, Prime[n]], With[{q = Quotient[k, 2]},
%t A340991      Sum[T[j, q] T[n - j, k - q], {j, 0, n}]]]];
%t A340991 Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 10 2021, after _Alois P. Heinz_ *)
%Y A340991 Columns k=0-4 give (offsets may differ): A000007, A000040, A014342, A014343, A014344.
%Y A340991 Main diagonal gives A000079.
%Y A340991 Row sums give A030017(n+1).
%Y A340991 T(2n,n) gives A340990.
%Y A340991 Cf. A030018, A030281.
%K A340991 nonn,tabl
%O A340991 0,3
%A A340991 _Alois P. Heinz_, Feb 01 2021