A243585 Expansion of x*log'(C(C(x)-1)-1), C(x) = (1-sqrt(1-4*x))/(2*x).

1, 4, 20, 106, 580, 3244, 18446, 106250, 618340, 3628600, 21438820, 127377980, 760346350, 4556473276, 27396081950, 165189725326, 998492094244, 6048338850560, 36706629690824, 223139239595840, 1358475322091620

Offset: 0

Author

Vladimir Kruchinin, Jun 07 2014

Keywords

• nonn

Links

• G.-S. Cheon, H. Kim, L. W. Shapiro, Mutation effects in ordered trees, arXiv preprint arXiv:1410.1249 [math.CO], 2014 (see page 6).

Crossrefs

Cf. A000108, A000984, A094527, A153294.

Programs

Mathematica

CoefficientList[Series[1/(Sqrt[(1-4*x)*(2*Sqrt[1-4*x]+5*x-2)/x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 08 2014 *)
A243585[n_] := Binomial[2 n, n] Hypergeometric2F1[1/2, -n, n + 1, -4];
Table[A243585[n], {n, 0, 20}] (* Peter Luschny, Aug 04 2019 *)

Maxima

a(n):=sum(binomial(2*k,k)*binomial(2*n,n-k),k,0,n);

Formula

a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(2*n,n-k).

a(n) ~ 5^(2*n+1/2) / (4^n * sqrt(3*Pi*n)). - Vaclav Kotesovec, Jun 08 2014

First column of A094527^2. 1 + x*exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 18*x^3 + 86*x^4 + ... is the o.g.f. for A153294. - Peter Bala, Jul 21 2015

Conjecture D-finite with recurrence: 2*n*(2*n-1)*(3*n-5)*a(n) +(-123*n^3+328*n^2-249*n+60)*a(n-1) +50*(n-1)*(2*n-3)*(3*n-2)*a(n-2)=0. - R. J. Mathar, Jun 14 2016

a(n) = binomial(2*n, n)*hypergeom([1/2, -n], [n + 1], -4). - Peter Luschny, Aug 04 2019