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

0,3

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

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

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 *)

Prog

(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
(Magma) m:=25; R<x>:=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

Crossrefs

Cf. A188314.

Sequence in context: A200539 A149369 A151432 * A006735 A166115 A027288

Adjacent sequences: A188309 A188310 A188311 * A188313 A188314 A188315

Keyword

nonn

Author

Paul Barry, Mar 28 2011

Status

approved