A188312 Expansion of (1/(1-x^2))*c(x/((1-x)*(1-x^2))) where c(x) is the g.f. of A000108.

1, 1, 4, 12, 45, 174, 709, 2978, 12825, 56303, 251060, 1133943, 5176926, 23851690, 110759081, 517853840, 2435786531, 11517940357, 54722081630, 261089977806, 1250479470053, 6009884614944, 28975052979797, 140098515402139, 679189779433800, 3300702453217325, 16076773046682690

Offset: 0

Author

Paul Barry, Mar 28 2011

Keywords

• nonn

Comments

Hankel transform is the (25,-29) Somos-4 sequence A188313. Image of Catalan numbers by A188316.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A188314.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x^2 -Sqrt(1-4*x-6*x^2+x^4))/(2*x*(1+x)))); // G. C. Greubel, Aug 14 2018

Maple

a := n -> add((-1)^(n-i)*hypergeom([(i+1)/2, i/2+1, i-n], [1, 2], 4), i=0..n);
seq(simplify(a(n)), n=0..26); # Peter Luschny, May 03 2018

Mathematica

CoefficientList[Series[(1-x^2 -Sqrt[1-4*x-6*x^2+x^4])/(2*x*(1+x)), {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *)

Maxima

a(n):=sum(sum((-1)^(n-k-i)*binomial(k+i-1, k-1)*binomial(2*k+i-2, k+i-1)* binomial(n-i-1, n-k-i)/k,k,1,n-i),i,0,n); /* Vladimir Kruchinin, May 03 2018 */

PARI

x='x+O('x^50); Vec((1-x^2 -sqrt(1-4*x-6*x^2+x^4))/(2*x*(1+x))) \\ G. C. Greubel, Aug 14 2018

Formula

G.f.: (1-x^2 - sqrt(1-4*x-6*x^2+x^4))/(2*x*(1+x)).

G.f.: u(x)=1/(1-x^2-x/(1-x-x*u(x))).

G.f.: 1/(1-x^2-x/(1-x-x/(1-x^2-x/(1-x-x/(1-...))))) (continued fraction).

Conjecture: (n+1)*a(n) +(3-4*n)*a(n-1) + (7-6*n)*a(n-2) -a(n-3) +(n-4)*a(n-4)=0. - R. J. Mathar, Nov 15 2011

a(n) = a(n-1) + (-1)^n + Sum_{i=0..n-1} a(i)*a(n-1-i). - Vladimir Kruchinin, May 03 2018

a(n) = Sum_{i=0..n} Sum_{k=1..n-i} (-1)^(n-k-i)*C(k+i-1,k-1)*C(2*k+i-2,k+i-1)*C(n-i-1,n-k-i)/k. - Vladimir Kruchinin, May 03 2018

a(n) = Sum_{i=0..n} (-1)^(n-i)*hypergeom([(i+1)/2, i/2+1, i-n], [1, 2], 4). - Peter Luschny, May 03 2018