A025246 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-3)*a(3) for n >= 4.

1, 0, 1, 1, 1, 2, 4, 7, 13, 26, 52, 104, 212, 438, 910, 1903, 4009, 8494, 18080, 38656, 82988, 178802, 386490, 837928, 1821664, 3970282, 8673258, 18987930, 41652382, 91539466, 201525238, 444379907, 981384125, 2170416738, 4806513660, 10657780276

Offset

1,6

Comments

Essentially the same as A023431.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 666

Formula

a(n) = A023431(n-3).

G.f.: (1+x-sqrt(1-2*x+x^2-4*x^3))/2. - Michael Somos, Jun 08 2000

n*a(n) = (2*n-3)*a(n-1) -(n-3)*a(n-2) +2*(2*n-9)*a(n-3). - R. J. Mathar, Feb 25 2015

a(n) = hypergeom([(3 - n)/3, (4 - n)/3, (5 - n)/3], [2, 3 - n], 27) for n >= 3. - Peter Luschny, Jun 15 2022

Maple

a := n -> ifelse(n < 3, 0^(n - 1),
   hypergeom([(3 - n)/3, (4 - n)/3, (5 - n)/3], [2, -n + 3], 27)):
seq(simplify(a(n)), n = 1..32); # Peter Luschny, Jun 15 2022

Mathematica

a[n_]:= a[n]= If[n<4, 1-Boole[n==2], Sum[a[j]*a[n-j], {j, n-3}]];
Table[a[n], {n, 45}] (* G. C. Greubel, Jun 15 2022 *)

Prog

(PARI) a(n)=polcoeff((1+x-sqrt(1-2*x+x^2-4*x^3+x*O(x^n)))/2, n)
(Magma) [n le 2 select 2-n else (&+[Binomial(n-k-3, 2*k)*Catalan(k): k in [0..Floor((n-3)/3)]]): n in [1..45]]; // G. C. Greubel, Jun 15 2022
(SageMath) [bool(n==1) + sum(binomial(n-k-3, 2*k)*catalan_number(k) for k in (0..((n-3)//3))) for n in (1..45)] // G. C. Greubel, Jun 15 2022

Crossrefs

Cf. A000108, A023431.

Sequence in context: A068031 A293314 A023431 * A256942 A112740 A309050

Adjacent sequences: A025243 A025244 A025245 * A025247 A025248 A025249

Keyword

nonn

Author

Clark Kimberling

Status

approved