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%I #15 May 23 2026 00:20:53
%S 1,1,1,1,1,1,1,1,1,1,2,0,1,2,1,1,3,0,1,3,1,1,1,4,1,0,1,5,2,0,1,6,0,0,
%T 1,7,2,1,1,9,1,0,1,10,3,0,1,12,2,0,1,14,3,1,1,1,17,1,0,0,1,19,5,1,0,1,
%U 23,3,0,0,1,26,5,2,0,1,30,4,0,0,1,35,7,0,0
%N Triangle read by rows: T(n, k) is the number of partitions of n into parts having exactly k prime factors (counted with multiplicity), with 0 <= k <= floor(log_2(n)) for n >= 1 and k = 0 for n = 0.
%H Felix Huber, <a href="/A394211/b394211.txt">Table of n, a(n) for n = 0..10009</a> (rows 0..1095, flattened)
%F G.f.: Sum_{n >= 0} T(n, k) x^n = Product_{m >= 1, A001222(m) = k} 1/(1 - x^m).
%F T(n, 0) = 1. For n >= 1 the only allowed part is 1; for n = 0 this counts the empty partition.
%F T(n, 1) = A000607(n).
%F T(n, 2) = A101048(n).
%F T(n, 3) = A112313(n).
%F T(n, 4) = A112314(n).
%F T(n, 5) = A112315(n).
%F T(n, 6) = A112316(n).
%F Sum_{k >= 0} T(n, k) = A319169(n).
%F For n >= 1, T(n, floor(log_2(n))) = 1 iff Omega(n) = floor(log_2(n)); otherwise 0.
%e The triangle begins:
%e n\k | 0 1 2 3
%e ----+---------------
%e 0 | 1
%e 1 | 1
%e 2 | 1 1
%e 3 | 1 1
%e 4 | 1 1 1
%e 5 | 1 2 0
%e 6 | 1 2 1
%e 7 | 1 3 0
%e 8 | 1 3 1 1
%e 9 | 1 4 1 0
%e 10 | 1 5 2 0
%e 11 | 1 6 0 0
%e 12 | 1 7 2 1
%e T(10, 2) = 2 since the two partitions [4, 6] and [10] both consist only of parts having exactly 2 prime factors (counted with multiplicity).
%p N := 22: # Enlarge if you want more rows
%p K := ilog2(N):
%p t := Array(0 .. N, 0 .. K):
%p for k from 0 to K do
%p t[0, k] := 1
%p end do:
%p v := Vector(N):
%p for i from 1 to N do
%p v[i] := NumberTheory:-Omega(i)
%p end do:
%p for k from 0 to K do
%p for i from 1 to N do
%p if v[i] = k then
%p for n from i to N do
%p t[n, k] := t[n, k] + t[n - i, k];
%p end do
%p end if
%p end do
%p end do:
%p T := (n, k) -> t[n, k]:
%p seq(seq(T(n, k), k = 0 .. ilog2(n)), n = 0 .. N);
%Y Cf. A000607, A001222, A101048, A112313, A112314, A112315, A112316, A319169.
%K nonn,tabf,easy
%O 0,11
%A _Felix Huber_, May 15 2026