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A274517 Number T(n,k) of integer partitions of n with exactly k distinct primes.

%I A274517 #16 Feb 28 2022 10:03:41
%S A274517 1,1,1,1,1,2,2,3,2,4,1,3,7,1,3,9,3,5,12,5,6,15,9,8,22,11,1,8,28,19,1,
%T A274517 12,38,24,3,13,46,38,4,17,62,48,8,19,77,68,12,26,98,87,20,28,117,127,
%U A274517 24,1,37,152,154,41,1,40,183,210,55,2,52,230,260,82,3
%N A274517 Number T(n,k) of integer partitions of n with exactly k distinct primes.
%C A274517 Row lengths increase by 1 at row A007504(n).
%C A274517 Columns k=0-1 give: A002095, A132381.
%C A274517 Row sums give: A000041.
%H A274517 Alois P. Heinz, <a href="/A274517/b274517.txt">Rows n = 0..1000, flattened</a>
%F A274517 G.f.: Product_{k>=1} (1 - x^prime(k))/(1 - x^k)*(y/(1-x^prime(k)) - y + 1).
%e A274517 T(6,1) = 7 because we have: 5+1, 4+2, 3+3, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1+1.
%e A274517 Triangle T(n,k) begins:
%e A274517   1;
%e A274517   1;
%e A274517   1,  1;
%e A274517   1,  2;
%e A274517   2,  3;
%e A274517   2,  4,  1;
%e A274517   3,  7,  1;
%e A274517   3,  9,  3;
%e A274517   5, 12,  5;
%e A274517   6, 15,  9;
%e A274517   8, 22, 11, 1;
%e A274517   ...
%p A274517 b:= proc(n, i) option remember; expand(
%p A274517       `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
%p A274517       `if`(j>0 and isprime(i), x, 1), j=0..n/i))))
%p A274517     end:
%p A274517 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A274517 seq(T(n), n=0..30);  # _Alois P. Heinz_, Jun 26 2016
%t A274517 nn = 20; Map[Select[#, # > 0 &] &, CoefficientList[Series[Product[
%t A274517       1/(1 - z^k), {k,Select[Range[1000], PrimeQ[#] == False &]}] Product[
%t A274517       u/(1 - z^j) - u + 1, {j, Table[Prime[n], {n, 1, nn}]}], {z, 0,
%t A274517      nn}], {z, u}]] // Grid
%Y A274517 Cf. A000041, A002095, A007504, A132381, A222656.
%K A274517 nonn,tabf
%O A274517 0,6
%A A274517 _Geoffrey Critzer_, Jun 25 2016