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

0,3

Links

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

Example

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;

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 *)

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.

Sequence in context: A379299 A011013 A138325 * A117175 A228086 A090482

Adjacent sequences: A258320 A258321 A258322 * A258324 A258325 A258326

Keyword

nonn,tabf

Author

Alois P. Heinz, May 26 2015

Status

approved