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A328775
Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} tau(n) * x^n, where tau = A000005.
3
1, 2, 0, 2, -1, 1, -1, 2, 1, -2, 0, 2, -1, -2, 2, 3, -2, -1, 2, 1, -4, 0, 3, -1, 3, -3, -2, 0, 1, 9, -15, 3, 17, -13, -1, -1, 9, -2, -18, 27, -10, -14, 24, -17, -15, 24, 27, -43, -37, 72, 43, -116, -11, 147, -98, -24, 67, -56, 24, -44, 213, -258, -193, 707, -435
OFFSET
1,2
COMMENTS
Inverse weigh transform of A000005.
LINKS
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; numtheory[tau](n)-b(n, n-1) end:
seq(a(n), n=1..80); # Alois P. Heinz, Oct 27 2019
MATHEMATICA
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] = DivisorSigma[0, n] - b[n, n - 1]; Array[a, 65]
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Oct 27 2019
STATUS
approved