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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. 20
 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=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-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 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 A321930 A134023 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 August 1 18:47 EDT 2021. Contains 346402 sequences. (Running on oeis4.)