A344447 - OEIS

%I #26 Aug 19 2021 04:47:02

%S 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,

%T 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,

%U 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

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

%C 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).

%H Alois P. Heinz, <a href="/A344447/b344447.txt">Rows n = 0..500, flattened</a>

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

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

%e Triangle T(n,k) begins:

%e   1 ;

%e   0 ;

%e   0 ;

%e   0 ;

%e   0, 1 ;

%e   0    ;

%e   0, 1 ;

%e   0    ;

%e   0, 0, 1 ;

%e   0, 1    ;

%e   0, 1, 1 ;

%e   0       ;

%e   0, 0, 1, 1 ;

%e   0, 0, 1    ;

%e   0, 1, 1, 1 ;

%e   0, 1, 1    ;

%e   0, 0, 1, 1, 1 ;

%e   0, 0, 0, 1    ;

%e   0, 0, 2, 2, 1 ;

%e   0, 0, 2, 1    ;

%e   0, 0, 2, 1, 1, 1 ;

%e   ...

%p h:= proc(n) option remember; `if`(n=0, 0,

%p      `if`(numtheory[bigomega](n)=2, n, h(n-1)))

%p     end:

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p      `if`(i>n, 0, expand(x*b(n-i, h(min(n-i, i)))))+b(n, h(i-1))))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n, h(n))):

%p seq(T(n), n=0..32);

%t h[n_] := h[n] = If[n == 0, 0,

%t      If[PrimeOmega[n] == 2, n, h[n-1]]];

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,

%t      If[i > n, 0, Expand[x*b[n-i, h[Min[n-i, i]]]]] + b[n, h[i-1]]]];

%t T[n_] := Table[Coefficient[#, x, i], {i, 0, Max[0, Exponent[#, x]]}]&[b[n, h[n]]];

%t Table[T[n], {n, 0, 32}] // Flatten (* _Jean-François Alcover_, Aug 19 2021, after _Alois P. Heinz_ *)

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

%Y Row sums give A101048.

%Y T(4n,n) gives A000012.

%Y Cf. A001358, A117278, A281617.

%K nonn,tabf

%O 0,45

%A _Alois P. Heinz_, May 19 2021