A319106 Expansion of Product_{k>=1} (1 + x^k)^ceiling(k/2).

1, 1, 1, 3, 4, 7, 11, 17, 26, 40, 60, 88, 131, 190, 276, 398, 568, 806, 1142, 1603, 2242, 3124, 4328, 5973, 8214, 11249, 15349, 20879, 28297, 38235, 51513, 69190, 92674, 123811, 164961, 219248, 290705, 384537, 507515, 668376, 878339, 1151899, 1507679, 1969503, 2567976, 3342227

Offset: 0

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

• nonn

Comments

Weigh transform of 1, 1, 2, 2, 3, 3, 4, 4, ... (A110654).

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A003293, A006054, A026007, A110654, A263140.

Programs

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}]

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