cp's OEIS Frontend

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

%I A319106 #9 Apr 03 2019 02:55:34
%S A319106 1,1,1,3,4,7,11,17,26,40,60,88,131,190,276,398,568,806,1142,1603,2242,
%T A319106 3124,4328,5973,8214,11249,15349,20879,28297,38235,51513,69190,92674,
%U A319106 123811,164961,219248,290705,384537,507515,668376,878339,1151899,1507679,1969503,2567976,3342227
%N A319106 Expansion of Product_{k>=1} (1 + x^k)^ceiling(k/2).
%C A319106 Weigh transform of 1, 1, 2, 2, 3, 3, 4, 4, ... (A110654).
%H A319106 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A319106 G.f.: Product_{k>=1} (1 + x^k)^A110654(k).
%F A319106 G.f.: Product_{k>=1} ((1 + x^(2*k-1))*(1 + x^(2*k)))^k.
%F A319106 G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*ceiling(d/2) ) * x^k/k).
%F A319106 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
%p A319106 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
%t A319106 nmax = 45; CoefficientList[Series[Product[(1 + x^k)^Ceiling[k/2], {k, 1, nmax}], {x, 0, nmax}], x]
%t A319106 nmax = 45; CoefficientList[Series[Product[((1 + x^(2 k - 1))(1 + x^(2 k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A319106 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}]
%Y A319106 Cf. A003293, A006054, A026007, A110654, A263140.
%K A319106 nonn
%O A319106 0,4
%A A319106 _Ilya Gutkovskiy_, Sep 10 2018