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A344257
Number of partitions of n into 10 semiprime parts.
2
1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 4, 3, 5, 3, 5, 5, 8, 8, 10, 8, 13, 13, 18, 17, 20, 19, 25, 28, 33, 33, 38, 40, 50, 52, 59, 63, 71, 75, 86, 94, 105, 110, 124, 131, 150, 159, 174, 189, 205, 217, 242, 264, 288, 303, 327, 354, 388, 414, 443, 476, 511, 547, 594, 641
OFFSET
40,7
FORMULA
a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} [Omega(r) = Omega(q) = Omega(p) = Omega(o) = Omega(m) = Omega(l) = Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k-l-m-o-p-q-r) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.
a(n) = [x^n y^10] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 19 2021
MAPLE
h:= proc(n) option remember; `if`(n=0, 0,
`if`(numtheory[bigomega](n)=2, n, h(n-1)))
end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
`if`(i>n, 0, x*b(n-i, h(min(n-i, i))))+b(n, h(i-1)))), x, 11)
end:
a:= n-> coeff(b(n, h(n)), x, 10):
seq(a(n), n=40..120); # Alois P. Heinz, May 26 2021
CROSSREFS
Cf. A001222 (Omega), A001358.
Column k=10 of A344447.
Sequence in context: A344254 A344255 A344256 * A088931 A088980 A254436
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, May 13 2021
EXTENSIONS
a(83)-a(103) from Alois P. Heinz, May 18 2021
STATUS
approved