A219180 Number T(n,k) of partitions of n into k distinct prime parts; triangle T(n,k), n>=0, read by rows.

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 1, 1, 1, 0, 0, 2, 2, 0, 0, 1, 2, 1, 0, 0, 2, 2, 0, 1, 0, 2, 2, 0, 0, 3, 2, 0, 0, 1, 2, 2, 0, 0, 2, 3, 1

Offset: 0

Author

Alois P. Heinz, Nov 13 2012

Keywords

• nonn
• look
• tabf

Comments

T(n,k) is defined for all n>=0 and k>=0. The triangle contains only elements with 0 <= k <= A024936(n). T(n,k) = 0 for k > A024936(n). Three rows are empty because there are no partitions of n into distinct prime parts for n in {1,4,6}.

Examples

			T(0,0) = 1: [], the empty partition.
T(2,1) = 1: [2].
T(5,1) = 1: [5], T(5,2) = 1: [2,3].
T(16,2) = 2: [5,11], [3,13].
Triangle T(n,k) begins:
  1;
  ;
  0, 1;
  0, 1;
  ;
  0, 1, 1;
  ;
  0, 1, 1;
  0, 0, 1;
  0, 0, 1;
  0, 0, 1, 1;
  0, 1;
  0, 0, 1, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..1000, flattened

Crossrefs

Columns k=0-10 give: A000007, A010051, A117929, A125688, A219198, A219199, A219200, A219201, A219202, A219203, A219204.

Row lengths are 1 + A024936(n).

Row sums give: A000586.

Last elements of rows give: A219181.

Row maxima give: A219182.

Least n with T(n,k) > 0 is A007504(k).

Cf. A000040, A117278, A219107.

Programs

Maple

b:= proc(n, i) option remember;
      `if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),
       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))
    end:
T:= proc(n) local l; l:= b(n, numtheory[pi](n));
       while nops(l)>0 and l[-1]=0 do l:= subsop(-1=NULL, l) od; l[]
    end:
seq(T(n), n=0..50);

Mathematica

nn=20;a=Table[Prime[n],{n,1,nn}];CoefficientList[Series[Product[1+y x^a[[i]],{i,1,nn}],{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, Nov 21 2012 *)
zip[f_, x_List, y_List, z_] := With[{m = Max[Length[x], Length[y]]}, f[PadRight[x, m, z], PadRight[y, m, z]]]; b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, zip[Plus, b[n, i-1], Join[{0}, If[Prime[i] > n, {}, b[n-Prime[i], i-1]]], 0]]]; T[n_] := Module[{l}, l = b[n, PrimePi[n]]; While[Length[l]>0 && l[[-1]] == 0, l = ReplacePart[l, -1 -> Sequence[]]]; l]; Table[T[n], {n, 0, 50}] // Flatten (* Jean-François Alcover, Jan 29 2014, after Alois P. Heinz *)

PARI

T(n)={ Vec(prod(k=1, n, 1 + isprime(k)*y*x^k + O(x*x^n))) }
{ my(t=T(20)); for(n=1, #t, print(if(t[n]!=0, Vecrev(t[n]), []))) } \\ Andrew Howroyd, Dec 22 2017

Formula

G.f. of column k: Sum_{0

T(n,k) = [x^n*y^k] Product_{i>=1} (1+x^prime(i)*y).

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