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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

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

Cf. A000010, A061255, A107742, A159929, A192065, A318975.

Sequence in context: A212549 A212550 A024786 * A097497 A332681 A279328

Adjacent sequences: A299066 A299067 A299068 * A299070 A299071 A299072

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 09 2018

Status

approved