A299069 Expansion of Product_{k>=1} (1 + x^k)^phi(k), where phi() is the Euler totient function (A000010).
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
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..5000
- N. J. A. Sloane, Transforms
Programs
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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}]
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