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A299069
Expansion of Product_{k>=1} (1 + x^k)^phi(k), where phi() is the Euler totient function (A000010).
12
1, 1, 1, 3, 4, 8, 11, 19, 30, 44, 69, 103, 157, 229, 341, 491, 722, 1038, 1488, 2128, 3015, 4267, 5989, 8407, 11713, 16289, 22523, 31097, 42729, 58569, 80003, 108957, 147983, 200383, 270693, 364631, 490105, 656961, 878775, 1172653, 1561626, 2074982, 2751648, 3641536, 4810009, 6341365, 8344967
OFFSET
0,4
LINKS
N. J. A. Sloane, Transforms
FORMULA
G.f.: Product_{k>=1} (1 + x^k)^A000010(k).
a(n) ~ exp(3^(5/3) * Zeta(3)^(1/3) * n^(2/3) / (2*Pi^(2/3))) * Zeta(3)^(1/6) / (2^(1/3) * 3^(1/6) * Pi^(5/6) * n^(2/3)). - Vaclav Kotesovec, Mar 23 2018
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
binomial(numtheory[phi](i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..50); # Alois P. Heinz, Mar 09 2018
MATHEMATICA
nmax = 46; CoefficientList[Series[Product[(1 + x^k)^EulerPhi[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d EulerPhi[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 46}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 09 2018
STATUS
approved