cp's OEIS Frontend

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

%I A344447 #26 Aug 19 2021 04:47:02
%S A344447 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 A344447 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 A344447 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 A344447 Number T(n,k) of partitions of n into k semiprimes; triangle T(n,k), n>=0, read by rows.
%C A344447 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 A344447 Alois P. Heinz, <a href="/A344447/b344447.txt">Rows n = 0..500, flattened</a>
%F A344447 T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A001358(j)).
%F A344447 Sum_{k>0} k * T(n,k) = A281617(n).
%e A344447 Triangle T(n,k) begins:
%e A344447   1 ;
%e A344447   0 ;
%e A344447   0 ;
%e A344447   0 ;
%e A344447   0, 1 ;
%e A344447   0    ;
%e A344447   0, 1 ;
%e A344447   0    ;
%e A344447   0, 0, 1 ;
%e A344447   0, 1    ;
%e A344447   0, 1, 1 ;
%e A344447   0       ;
%e A344447   0, 0, 1, 1 ;
%e A344447   0, 0, 1    ;
%e A344447   0, 1, 1, 1 ;
%e A344447   0, 1, 1    ;
%e A344447   0, 0, 1, 1, 1 ;
%e A344447   0, 0, 0, 1    ;
%e A344447   0, 0, 2, 2, 1 ;
%e A344447   0, 0, 2, 1    ;
%e A344447   0, 0, 2, 1, 1, 1 ;
%e A344447   ...
%p A344447 h:= proc(n) option remember; `if`(n=0, 0,
%p A344447      `if`(numtheory[bigomega](n)=2, n, h(n-1)))
%p A344447     end:
%p A344447 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A344447      `if`(i>n, 0, expand(x*b(n-i, h(min(n-i, i)))))+b(n, h(i-1))))
%p A344447     end:
%p A344447 T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n, h(n))):
%p A344447 seq(T(n), n=0..32);
%t A344447 h[n_] := h[n] = If[n == 0, 0,
%t A344447      If[PrimeOmega[n] == 2, n, h[n-1]]];
%t A344447 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
%t A344447      If[i > n, 0, Expand[x*b[n-i, h[Min[n-i, i]]]]] + b[n, h[i-1]]]];
%t A344447 T[n_] := Table[Coefficient[#, x, i], {i, 0, Max[0, Exponent[#, x]]}]&[b[n, h[n]]];
%t A344447 Table[T[n], {n, 0, 32}] // Flatten (* _Jean-François Alcover_, Aug 19 2021, after _Alois P. Heinz_ *)
%Y A344447 Columns k=0-10 give: A000007, A064911, A072931, A344446, A340756, A344245, A344246, A344254, A344255, A344256, A344257.
%Y A344447 Row sums give A101048.
%Y A344447 T(4n,n) gives A000012.
%Y A344447 Cf. A001358, A117278, A281617.
%K A344447 nonn,tabf
%O A344447 0,45
%A A344447 _Alois P. Heinz_, May 19 2021