A117278 Triangle read by rows: T(n,k) is the number of partitions of n into k prime parts (n>=2, 1<=k<=floor(n/2)).

1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1, 1, 1, 1, 0, 2, 2, 1, 0, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 0, 2, 1, 3, 2, 1, 1, 0, 1, 3, 2, 3, 2, 1, 0, 2, 2, 3, 3, 2, 1, 1, 1, 0, 4, 3, 3, 3, 2, 1, 0, 2, 2, 4, 3, 4, 2, 1, 1, 1, 1, 3, 4, 5, 3, 3, 2, 1, 0, 2, 2, 6, 4, 4, 4, 2, 1, 1, 0, 1, 5, 3, 6

Offset

2,19

Comments

Row n has floor(n/2) terms. Row sums yield A000607. T(n,1) = A010051(n) (the characteristic function of the primes). T(n,2) = A061358(n). Sum(k*T(n,k), k>=1) = A084993(n).

Links

Alois P. Heinz, Rows n = 2..200, flattened

Formula

G.f.: G(t,x) = -1+1/product(1-tx^(p(j)), j=1..infinity), where p(j) is the j-th prime.

Example

T(12,3) = 2 because we have [7,3,2] and [5,5,2].
Triangle starts:
  1;
  1;
  0, 1;
  1, 1;
  0, 1, 1;
  1, 1, 1;
  0, 1, 1, 1;
  0, 1, 2, 1;
  ...

Maple

g:=1/product(1-t*x^(ithprime(j)), j=1..30): gser:=simplify(series(g, x=0, 30)): for n from 2 to 22 do P[n]:=sort(coeff(gser, x^n)) od: for n from 2 to 22 do seq(coeff(P[n], t^j), j=1..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
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))[]], 0)))
    end:
T:= n-> subsop(1=NULL, b(n, numtheory[pi](n)))[]:
seq(T(n), n=2..25);  # Alois P. Heinz, Nov 16 2012

Mathematica

(* As triangle: *) nn=20; a=Product[1/(1-y x^i), {i, Table[Prime[n], {n, 1, nn}]}]; Drop[CoefficientList[Series[a, {x, 0, nn}], {x, y}], 2, 1]//Grid (* Geoffrey Critzer, Oct 30 2012 *)

Prog

(PARI)
parts(n, pred)={prod(k=1, n, if(pred(k), 1/(1-y*x^k) + O(x*x^n), 1))}
{my(n=15); apply(p->Vecrev(p/y), Vec(parts(n, isprime)-1))} \\ Andrew Howroyd, Dec 28 2017

Crossrefs

Columns k=1-10 give: A010051, A061358, A068307, A259194, A259195, A259196, A259197, A259198, A259200, A259201.

Row sums give A000607.

T(A000040(n),n) gives A259254(n).

Cf. A084993, A219180.

Sequence in context: A030187 A270657 A270658 * A215556 A140082 A330714

Adjacent sequences: A117275 A117276 A117277 * A117279 A117280 A117281

Keyword

nonn,tabf

Author

Emeric Deutsch, Mar 07 2006

Status

approved