A025230 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3.

3, 1, 6, 37, 234, 1514, 9996, 67181, 458562, 3172478, 22206420, 157027938, 1120292388, 8055001716, 58314533400, 424740506109, 3110401363122, 22888001498102, 169155516667524, 1255072594261142, 9345400450314924

Offset

1,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.

Formula

G.f.: (1-sqrt(1-12*x+32*x^2))/2. - Michael Somos, Jun 08 2000

D-finite with recurrence n*a(n) = (12*n-18)*a(n-1) - 32*(n-3)*a(n-2) - Richard Choulet, Dec 17 2009

a(n) ~ 2^(3*n-5/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 11 2013

a(n) = 4^(n-2)*hypergeom([3/2, -n+2], [3], -1) for n>1. - Peter Luschny, Feb 03 2015

a(n+1) = GegenbauerC(n-1, -n, -3)/n for n>=1. - Peter Luschny, May 09 2016

From Peter Bala, Feb 03 2024: (Start)

G.f: 3*x + x^2/(1 - 4*x) * c(x/(1 - 4*x))^2, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.

a(n+2) = Sum_{k = 0..n} 4^(n-k)*binomial(n, k)*Catalan(k+1).

G.f: 3*x + x^2/(1 - 8*x) * c(-x/(1 - 8*x))^2.

a(n+2) = 8^n * Sum_{k = 0..n} (-8)^(-k)*binomial(n, k)*Catalan(k+1).

a(n+2) = 8^n * hypergeom([-n, 3/2], [3], 1/2).

a(n) is odd iff n is a power of 2. (End)

Maple

h := n -> simplify(4^n*hypergeom([3/2, -n], [3], -1)):
a := n -> `if`(n=1, 3, h(n-2)):
seq(a(n), n=1..21); # Peter Luschny, Feb 03 2015

Mathematica

Rest[CoefficientList[Series[(1-Sqrt[1-12x+32x^2])/2, {x, 0, 30}], x]]  (* Harvey P. Dale, Feb 22 2011 *)

Prog

(PARI) a(n)=polcoeff((1-sqrt(1-12*x+32*x^2+x*O(x^n)))/2, n)
(PARI) {a(n)=if(n<2, 3*(n==1), n--; polcoeff( serreverse( x/(1+6*x+x^2) +x*O(x^n) ), n))} /* Michael Somos, Oct 14 2006 */

Crossrefs

For Sum_{k = 0..n} m^(n-k)*binomial(n, k)*Catalan(k+1) see A126120 (m = -2), A001006 (m = -1), A000108 (m = 0), A002212 (m = 1), A005572 (m = 2), A182401 (m = 3), A025230 (m = 4).

Sequence in context: A157866 A221852 A363196 * A152456 A128605 A051511

Adjacent sequences: A025227 A025228 A025229 * A025231 A025232 A025233

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved