OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..120 from Robert Israel)
FORMULA
a(n) = Sum_{k=0..n} binomial(n+(k)-1,(k))*binomial(3*n-1,n-4*k).
a(n) ~ sqrt((6+2*sqrt(2))/7) * (7+5*sqrt(2))^n / (sqrt(Pi*n) * 2^(n+5/4)). - Vaclav Kotesovec, Jun 15 2014
a(n) = [x^n] ( (1 - x)^2/((1 - x)^4 - x^4) )^n. O.g.f. A(x) is essentially obtained by logarithmically differentiating the o.g.f of A243632. - Peter Bala, Oct 02 2015
MATHEMATICA
CoefficientList[Series[-((2*(5 + Sqrt[1-4*x] + Sqrt[-6+10*Sqrt[1-4*x] - 4*x]) * x)/((-3 + Sqrt[1-4*x] + Sqrt[-6 + 10*Sqrt[1-4*x] - 4*x]) * Sqrt[1-4*x]*Sqrt[-6 + 10*Sqrt[1-4*x] - 4*x])), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 15 2014 *)
Table[Sum[Binomial[n + k - 1, k]*Binomial[3*n - 1, n - 4*k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Feb 14 2017 *)
PROG
(Maxima)
a(n):=sum(binomial(n+(k)-1, (k))*binomial(3*n-1, n-4*k), k, 0, n);
(PARI) a(n) = sum(k=0, n, binomial(n+k-1, (k))*binomial(3*n-1, n-4*k));
vector(20, n, a(n-1)) \\ Altug Alkan, Oct 02 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Jun 08 2014
STATUS
approved