A219180 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,41

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}.

LINKS

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

FORMULA

G.f. of column k: Sum_{0<i_1<i_2<...<i_k} x^(Sum_{j=1..k} prime(i_j)).

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

EXAMPLE

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:

;

0, 1;

;

0, 1, 1;

;

0, 1, 1;

0, 0, 1;

0, 0, 1, 1;

0, 1;

0, 0, 1, 1;

...

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 *)

PROG

(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

CROSSREFS