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A282516 Number T(n,k) of k-element subsets of [n] having a prime element sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A282516 #18 Oct 22 2023 13:08:13
%S A282516 0,0,0,0,1,1,0,2,2,0,0,2,4,1,0,0,3,5,2,2,0,0,3,7,6,4,2,0,0,4,9,10,11,
%T A282516 7,1,0,0,4,11,18,21,13,7,2,0,0,4,14,26,34,31,20,7,3,0,0,4,18,37,53,59,
%U A282516 51,32,11,2,0,0,5,21,47,82,110,117,85,35,12,2,0
%N A282516 Number T(n,k) of k-element subsets of [n] having a prime element sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A282516 Alois P. Heinz, <a href="/A282516/b282516.txt">Rows n = 0..200, flattened</a>
%e A282516 Triangle T(n,k) begins:
%e A282516   0;
%e A282516   0, 0;
%e A282516   0, 1,  1;
%e A282516   0, 2,  2,  0;
%e A282516   0, 2,  4,  1,  0;
%e A282516   0, 3,  5,  2,  2,   0;
%e A282516   0, 3,  7,  6,  4,   2,   0;
%e A282516   0, 4,  9, 10, 11,   7,   1,  0;
%e A282516   0, 4, 11, 18, 21,  13,   7,  2,  0;
%e A282516   0, 4, 14, 26, 34,  31,  20,  7,  3,  0;
%e A282516   0, 4, 18, 37, 53,  59,  51, 32, 11,  2, 0;
%e A282516   0, 5, 21, 47, 82, 110, 117, 85, 35, 12, 2, 0;
%e A282516   ...
%p A282516 b:= proc(n, s) option remember; expand(`if`(n=0,
%p A282516       `if`(isprime(s), 1, 0), b(n-1, s)+x*b(n-1, s+n)))
%p A282516     end:
%p A282516 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
%p A282516 seq(T(n), n=0..16);
%t A282516 b[n_, s_] := b[n, s] = Expand[If[n==0, If[PrimeQ[s], 1, 0], b[n-1, s] + x*b[n-1, s+n]]];
%t A282516 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
%t A282516 Table[T[n], {n, 0, 16}] // Flatten (* _Jean-François Alcover_, Mar 21 2017, translated from Maple *)
%Y A282516 Columns k=0-10 give: A000004, A000720, A071917, A320678, A320679, A320680, A320681, A320682, A320683, A320684, A320685.
%Y A282516 Row sums give A127542.
%Y A282516 Main diagonal gives A185012.
%Y A282516 First lower diagonal gives A282518.
%Y A282516 T(2n,n) gives A282517.
%K A282516 nonn,tabl
%O A282516 0,8
%A A282516 _Alois P. Heinz_, Feb 17 2017