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