A344447 Number T(n,k) of partitions of n into k semiprimes; triangle T(n,k), n>=0, read by rows.

1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 2, 2, 1, 0, 0, 2, 1, 0, 0, 2, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 3, 2, 1, 0, 0, 1, 2, 1, 0, 0, 2, 3, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 1, 2, 3, 2, 1, 0, 0, 1, 3, 3, 1, 0, 0, 2, 3, 4, 2, 1, 1

Offset

0,45

Comments

T(n,k) is defined for all n,k >= 0. The triangle contains in each row n only the terms for k=0 and then up to the last positive T(n,k) (if it exists).

Links

Alois P. Heinz, Rows n = 0..500, flattened

Formula

T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A001358(j)).

Sum_{k>0} k * T(n,k) = A281617(n).

Example

Triangle T(n,k) begins:
  1 ;
  0 ;
  0 ;
  0 ;
  0, 1 ;
  0    ;
  0, 1 ;
  0    ;
  0, 0, 1 ;
  0, 1    ;
  0, 1, 1 ;
  0       ;
  0, 0, 1, 1 ;
  0, 0, 1    ;
  0, 1, 1, 1 ;
  0, 1, 1    ;
  0, 0, 1, 1, 1 ;
  0, 0, 0, 1    ;
  0, 0, 2, 2, 1 ;
  0, 0, 2, 1    ;
  0, 0, 2, 1, 1, 1 ;
  ...

Maple

h:= proc(n) option remember; `if`(n=0, 0,
     `if`(numtheory[bigomega](n)=2, n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
     `if`(i>n, 0, expand(x*b(n-i, h(min(n-i, i)))))+b(n, h(i-1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n, h(n))):
seq(T(n), n=0..32);

Mathematica

h[n_] := h[n] = If[n == 0, 0,
     If[PrimeOmega[n] == 2, n, h[n-1]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     If[i > n, 0, Expand[x*b[n-i, h[Min[n-i, i]]]]] + b[n, h[i-1]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, Max[0, Exponent[#, x]]}]&[b[n, h[n]]];
Table[T[n], {n, 0, 32}] // Flatten (* Jean-François Alcover, Aug 19 2021, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A000007, A064911, A072931, A344446, A340756, A344245, A344246, A344254, A344255, A344256, A344257.

Row sums give A101048.

T(4n,n) gives A000012.

Cf. A001358, A117278, A281617.

Sequence in context: A335833 A258747 A160806 * A287385 A191411 A133418

Adjacent sequences: A344444 A344445 A344446 * A344448 A344449 A344450

Keyword

nonn,tabf

Author

Alois P. Heinz, May 19 2021

Status

approved