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A219180 Number T(n,k) of partitions of n into k distinct prime parts; triangle T(n,k), n>=0, read by rows. 18
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 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:

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;

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-Fran├žois Alcover, Jan 29 2014, after Alois P. Heinz *)

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.

Sequence in context: A085252 A250214 A073423 * A179952 A134023 A257931

Adjacent sequences:  A219177 A219178 A219179 * A219181 A219182 A219183

KEYWORD

nonn,look,tabf

AUTHOR

Alois P. Heinz, Nov 13 2012

STATUS

approved

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Last modified June 24 07:19 EDT 2017. Contains 288697 sequences.