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A353161
Product_{n>=1} (1 + x^n)^a(n) = 1 + x + Sum_{n>=2} prime(n-1) * x^n.
1
1, 2, 1, 3, 1, 2, -1, 1, -3, -1, 4, 1, 5, 2, -4, -4, -9, 0, -3, 14, 19, 4, 6, -38, -27, -17, 5, 59, 50, 103, -49, -100, -142, -222, 83, 138, 468, 362, 0, -313, -1215, -599, -526, 961, 2572, 1837, 1673, -2858, -4516, -6182, -3880, 5981, 9282, 18218, 7414, -8554, -24446
OFFSET
1,2
COMMENTS
Inverse weigh transform of {1, primes}.
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= proc(n) option remember; `if`(n=1, 1, ithprime(n-1))-b(n, n-1) end:
seq(a(n), n=1..60); # Alois P. Heinz, Apr 28 2022
MATHEMATICA
p[n_] := If[n == 1, 1, Prime[n - 1]]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = p[n] - b[n, n - 1]; Table[a[n], {n, 1, 57}]
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Apr 28 2022
STATUS
approved