A281244 Expansion of Product_{k>=1} (1 + x^(6*k-1)).

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 2, 3, 1, 0, 0, 0, 3, 4, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 5, 5, 1, 0, 0, 2, 7, 5, 1, 0, 0, 3, 8, 6, 1, 0, 0, 5, 10, 6, 1, 0, 1

Offset: 0

Author

Vaclav Kotesovec, Jan 18 2017

Keywords

• nonn

Comments

Convolution of this sequence and A280456 is A098884.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Vaclav Kotesovec, Graph - minor asymptotic term, (Pi/144 - 9/(4*Pi))/sqrt(2) = -0.49100125...

Crossrefs

Cf. A262928, A147599, A281243, A281245.

Cf. A261612, A169975, A280454, A280456, A280457.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(add(
      [0$5, 1, 0$4, -1, 1][1+irem(d, 12)]*d, d=
       numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 18 2017

Mathematica

nmax = 200; CoefficientList[Series[Product[(1 + x^(6*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 6] == 5, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

Formula

a(n) ~ exp(sqrt(n/2)*Pi/3) / (2^(25/12)*sqrt(3)*n^(3/4)) * (1 + (Pi/144 - 9/(4*Pi)) / sqrt(2*n)).

G.f.: Sum_{k>=0} x^(k*(3*k + 2)) / Product_{j=1..k} (1 - x^(6*j)). - Ilya Gutkovskiy, Nov 24 2020