A277969 a(n) = Sum_{k=0..n} binomial(n-3,n-k)*Catalan(k).

1, -1, 2, 5, 19, 75, 305, 1270, 5390, 23236, 101480, 448085, 1997115, 8973255, 40602093, 184853055, 846206025, 3892585325, 17984308775, 83417287855, 388297304825, 1813341109825, 8493372326675, 39889629750600, 187812852106636

Offset: 0

Author

Vladimir Kruchinin, Nov 06 2016

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 0..1430

Crossrefs

Cf. A000108, A055452.

Programs

Maple

f:= gfun:-rectoproc({(5*n-10)*a(n)+(-7-6*n)*a(n+1)+(n+3)*a(n+2),a(0) = 1, a(1) = -1, a(2) = 2, a(3) = 5},a(n),remember):
map(f, [$0..30]); # Robert Israel, Nov 21 2016

Mathematica

CoefficientList[Series[((1 - x)^3 (1 - Sqrt[(5 x - 1) / (x - 1)])) / (2 x), {x, 0, 25}], x] (* Vincenzo Librandi, Nov 07 2016 *)

Maxima

a(n):=sum((binomial(2*k,k)*binomial(n-3,n-k))/(k+1),k,0,n);

PARI

x='x+O('x^50); Vec(((1-x)^3*(1-sqrt((5*x-1)/(x-1))))/(2*x)) \\ G. C. Greubel, Apr 09 2017

Formula

G.f.: ((1-x)^3*(1-sqrt((5*x-1)/(x-1))))/(2*x).

a(n) ~ 8*5^(n-3/2) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 07 2016

(5*n-10)*a(n)-(7+6*n)*a(n+1)+(n+3)*a(n+2)=0 for n >= 2. - Robert Israel, Nov 21 2016

a(n) = A055452(n+1) for n > 2. - Georg Fischer, Oct 23 2018