A258323 Sum T(n,k) over all partitions lambda of n into k distinct parts of Product_{i:lambda} prime(i); triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

1, 0, 2, 0, 3, 0, 5, 6, 0, 7, 10, 0, 11, 29, 0, 13, 43, 30, 0, 17, 94, 42, 0, 19, 128, 136, 0, 23, 231, 293, 0, 29, 279, 551, 210, 0, 31, 484, 892, 330, 0, 37, 584, 1765, 852, 0, 41, 903, 2570, 1826, 0, 43, 1051, 4273, 4207, 0, 47, 1552, 6747, 6595, 2310

Offset: 0

Author

Alois P. Heinz, May 26 2015

Keywords

• nonn
• tabf

Examples

			T(6,2) = 43 because the partitions of 6 into 2 distinct parts are {[5,1], [4,2]} and prime(5)*prime(1) + prime(4)*prime(2) = 11*2 + 7*3 = 22 + 21 = 43.
Triangle T(n,k) begins:
  1
  0,  2;
  0,  3;
  0,  5,   6;
  0,  7,  10;
  0, 11,  29;
  0, 13,  43,  30;
  0, 17,  94,  42;
  0, 19, 128, 136;
  0, 23, 231, 293;
  0, 29, 279, 551, 210;
		

Links

• Alois P. Heinz, Rows n = 0..500, flattened

Crossrefs

Columns k=0-10 give: A000007, A000040, A025129(n+1), A258358, A258359, A258360, A258361, A258362, A258363, A258364, A258365.

Row sums give A147655.

T(n*(n+1)/2,n) = A002110(n).

T(n^2,n) = A321267(n).

Cf. A000217, A003056, A145518, A246867.

Programs

Maple

g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, expand(
      add(g(n-i*j, i-1)*(ithprime(i)*x)^j, j=0..min(1, n/i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n$2)):
seq(T(n), n=0..20);

Mathematica

g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, Expand[Sum[g[n-i*j, i-1] * (Prime[i]*x)^j, {j, 0, Min[1, n/i]}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][g[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 06 2017, after Alois P. Heinz *)