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

1, 1, 2, 8, 27, 103, 389, 1497, 5786, 22556, 88230, 346576, 1365119, 5390585, 21327913, 84527939, 335477433, 1333079925, 5302763618, 21112688376, 84125853415, 335443149005, 1338370995240, 5342843332758, 21339341267983, 85266832981905, 340840044333836, 1362936812554758

Offset: 0

Author

Ilya Gutkovskiy, May 28 2018

Keywords

• nonn

Comments

Weigh transform of the central binomial coefficients 1, 2, 6, 20, 70, ... (A000984).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000984, A194353.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(binomial(2*i-2, i-1), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 28 2018

Mathematica

nmax = 27; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) x^k/(k Sqrt[1 - 4 x^k]), {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 27; CoefficientList[Series[Product[(1 + x^k)^Binomial[2 k - 2, k - 1], {k, 1, nmax}], {x, 0, nmax}], x]
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}]

Formula

G.f.: Product_{k>=1} (1 + x^k)^binomial(2*k-2,k-1).