A270363 a(n) = (n+1)*Sum_{k=0..(n-1)/2}((binomial(2*n-3*k-2,n-k-1))/(n-k)).

0, 2, 3, 10, 30, 101, 350, 1250, 4548, 16782, 62579, 235273, 890331, 3387204, 12943353, 49643762, 191010623, 736946570, 2850013623, 11044973890, 42882986660, 166770990377, 649526893537, 2533096497017, 9890766366030, 38662031939117

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

G.f.: (1-sqrt(1-4*x))/(sqrt(1-4*x)*x-x+2)*((6*x+sqrt(1-4*x)-1)/(4*x+sqrt(1-4*x)-1)).

Conjecture: n*(7*n^2-17*n-2) *a(n) +(-35*n^3+99*n^2+20*n-120) *a(n-1) +2* (2*n-5) *(7*n^2-3*n-12)*a(n-2) +n*(7*n^2-17*n-2) *a(n-3) +2*-(2*n-5) *(7*n^2-3*n-12) *a(n-4)=0. - R. J. Mathar, Mar 22 2016

a(n) ~ 2^(2*n+1) / (7*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 22 2016

Mathematica

CoefficientList[Series[(1 - Sqrt[1 - 4*x])/(Sqrt[1 - 4*x]*x - x + 2)* ((6*x + Sqrt[1 - 4*x] - 1)/(4*x + Sqrt[1 - 4*x] - 1)), {x, 0, 50}], x] (* G. C. Greubel, Jun 04 2017 *)

Prog

(Maxima)
taylor((1-sqrt(1-4*x))/(sqrt(1-4*x)*x-x+2)*((6*x+sqrt(1-4*x)-1)/(4*x+sqrt(1-4*x)-1)), x, 0, 15);
a(n):=(n+1)*sum((binomial(2*n-3*k-2, n-k-1))/(n-k), k, 0, (n-1)/2);
(PARI) x='x+O('x^100); concat(0, Vec((1-sqrt(1-4*x))/(sqrt(1-4*x)*x-x+2)*((6*x+sqrt(1-4*x)-1)/(4*x+sqrt(1-4*x)-1)))) \\ Altug Alkan, Mar 25 2016

Crossrefs

Cf. A000108, A033184.

Sequence in context: A301971 A319671 A338593 * A131764 A066706 A353052

Adjacent sequences: A270360 A270361 A270362 * A270364 A270365 A270366

Keyword

nonn

Author

Vladimir Kruchinin, Mar 22 2016

Status

approved