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A219180
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Number T(n,k) of partitions of n into k distinct prime parts; triangle T(n,k), n>=0, read by rows.
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24
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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
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OFFSET
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0,41
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COMMENTS
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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}.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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;
...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Columns k=0-10 give: A000007, A010051, A117929, A125688, A219198, A219199, A219200, A219201, A219202, A219203, A219204.
Last elements of rows give: A219181.
Least n with T(n,k) > 0 is A007504(k).
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KEYWORD
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AUTHOR
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STATUS
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approved
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