A025230 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 3,1.
3, 1, 6, 37, 234, 1514, 9996, 67181, 458562, 3172478, 22206420, 157027938, 1120292388, 8055001716, 58314533400, 424740506109, 3110401363122, 22888001498102, 169155516667524, 1255072594261142, 9345400450314924, 69812926066668044, 523072984217339304
Offset: 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.
Crossrefs
Programs
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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
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Mathematica
Rest[CoefficientList[Series[(1-Sqrt[1-12x+32x^2])/2,{x,0,30}],x]] (* Harvey P. Dale, Feb 22 2011 *) -
PARI
a(n)=polcoeff((1-sqrt(1-12*x+32*x^2+x*O(x^n)))/2,n)
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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 */
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)
Extensions
Name clarified by Robert C. Lyons, Feb 06 2025