OFFSET
1,4
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
FORMULA
G.f.: x*(3*x*C(x)-1)/((1-2*x*C(x))*(1-x*C(x))^2), where C(x) is the g.f. of A000108.
a(n) = Sum_{k=1..n} k*(-1)^k*binomial(n-1, k-1)*binomial(3*n-k-1, n-k))/n.
a(n) = (2^(2*n-1)*(n-3)*(n-1/2)!)/(sqrt(Pi)*(n+1)!). - Peter Luschny, Nov 14 2014
D-finite with recurrence (for n > 4): (n-4)*(n+1)*a(n) = 2*(n-3)*(2*n-1)*a(n-1). - Vaclav Kotesovec, Nov 14 2014
G.f.: x^4* 2F1(2,9/2;6;4*x) -x -x^2. - R. J. Mathar, Jan 25 2020
From Peter Bala, Feb 13 2024: (Start)
a(n) = 2*binomial(2*n-1, n+1) - binomial(2*n-1, n) = binomial(2*n-1, n+1) - Catalan(n).
a(n) is odd iff n = 2^k for k >= 0. (End)
MATHEMATICA
Table[Sum[k*(-1)^k*Binomial[n-1, k-1]*Binomial[3*n-k-1, n-k], {k, 1, n}]/n, {n, 1, 20}] (* Vaclav Kotesovec, Nov 14 2014 after Vladimir Kruchinin *)
Rest[Rest[CoefficientList[Series[1-(7*x-2)/(2*Sqrt[1-4*x]), {x, 0, 30}], x]]] (* Vaclav Kotesovec, Nov 14 2014 *)
PROG
(Maxima)
a(n):=sum(k*(-1)^k*binomial(n-1, k-1)*binomial(3*n-k-1, n-k), k, 1, n)/n;
(Sage)
a = lambda n: (2^(2*n-1)*(n-3)*factorial(n-1/2))/(sqrt(pi)* factorial(n+1))
[a(n) for n in (1..20)] # Peter Luschny, Nov 14 2014
(PARI) x='x+O('x^50); Vec((3*x/2-1-(7*x-2)/(2*sqrt(1-4*x)))/x) \\ G. C. Greubel, Jun 02 2017
CROSSREFS
KEYWORD
sign
AUTHOR
Vladimir Kruchinin, Nov 14 2014
STATUS
approved