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A305256 Expansion of exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*sqrt(1 - 4*x^k))).

%I A305256 #12 Jun 09 2018 02:21:55
%S A305256 1,1,2,8,27,103,389,1497,5786,22556,88230,346576,1365119,5390585,
%T A305256 21327913,84527939,335477433,1333079925,5302763618,21112688376,
%U A305256 84125853415,335443149005,1338370995240,5342843332758,21339341267983,85266832981905,340840044333836,1362936812554758
%N A305256 Expansion of exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*sqrt(1 - 4*x^k))).
%C A305256 Weigh transform of the central binomial coefficients 1, 2, 6, 20, 70, ... (A000984).
%H A305256 Alois P. Heinz, <a href="/A305256/b305256.txt">Table of n, a(n) for n = 0..1000</a>
%H A305256 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A305256 G.f.: Product_{k>=1} (1 + x^k)^binomial(2*k-2,k-1).
%p A305256 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A305256       binomial(binomial(2*i-2, i-1), j)*b(n-i*j, i-1), j=0..n/i)))
%p A305256     end:
%p A305256 a:= n-> b(n$2):
%p A305256 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 28 2018
%t A305256 nmax = 27; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) x^k/(k Sqrt[1 - 4 x^k]), {k, 1, nmax}]], {x, 0, nmax}], x]
%t A305256 nmax = 27; CoefficientList[Series[Product[(1 + x^k)^Binomial[2 k - 2, k - 1], {k, 1, nmax}], {x, 0, nmax}], x]
%t A305256 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d Binomial[2 d - 2, d - 1], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]
%Y A305256 Cf. A000984, A194353.
%K A305256 nonn
%O A305256 0,3
%A A305256 _Ilya Gutkovskiy_, May 28 2018