OFFSET
0,4
COMMENTS
Weigh transform of 1, 1, 2, 2, 3, 3, 4, 4, ... (A110654).
LINKS
N. J. A. Sloane, Transforms
FORMULA
G.f.: Product_{k>=1} (1 + x^k)^A110654(k).
G.f.: Product_{k>=1} ((1 + x^(2*k-1))*(1 + x^(2*k)))^k.
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*ceiling(d/2) ) * x^k/k).
a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(5/3) + Pi^2 * n^(1/3) / (2^(10/3) * 3^(4/3) * Zeta(3)^(1/3)) - Pi^4 / (2^7 * 3^4 * Zeta(3))) * Zeta(3)^(1/6) / (2^(7/8) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Sep 11 2018
MAPLE
a:=series(mul((1+x^k)^ceil(k/2), k=1..100), x=0, 46): seq(coeff(a, x, n), n=0..45); # Paolo P. Lava, Apr 02 2019
MATHEMATICA
nmax = 45; CoefficientList[Series[Product[(1 + x^k)^Ceiling[k/2], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Product[((1 + x^(2 k - 1))(1 + x^(2 k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d Ceiling[d/2], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 45}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 10 2018
STATUS
approved