A277957 a(n) = (n+3)*Sum_{i=0..n} binomial(3*n-2*i+2,n-i)/(2*n-i+3).

1, 5, 26, 144, 834, 4979, 30361, 188003, 1177694, 7443721, 47384897, 303389530, 1951806313, 12607088771, 81709809546, 531138264252, 3461366814726, 22607751250442, 147952881721126, 969953549401499, 6368831275489633

Offset

0,2

Links

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

Formula

G.f.: F'(x)*F(x)^2/(1-F(x))/x^2, where F(x)/x is the g.f. of A001764.

From Vaclav Kotesovec, Nov 06 2016: (Start)

Recurrence: 2*(n+1)*(2*n + 3)*(91*n^4 - 24*n^3 - 313*n^2 + 246*n - 24)*a(n) = (2821*n^6 + 5080*n^5 - 12775*n^4 - 5200*n^3 + 8454*n^2 + 420*n - 720)*a(n-1) - (2821*n^6 + 5080*n^5 - 12775*n^4 - 5200*n^3 + 8454*n^2 + 420*n - 720)*a(n-2) + 3*(3*n - 4)*(3*n - 2)*(91*n^4 + 340*n^3 + 161*n^2 - 88*n - 24)*a(n-3).

a(n) ~ 3^(3*n+9/2) / (7 * sqrt(Pi*n) * 2^(2*n+4)). (End)

a(n) = A262394(n-1)*hypergeom([1,-2*n-3,-n],[-3*n/2-1,-3*n/2-1/2],1/4). - Peter Luschny, Nov 06 2016

Maple

h := n -> hypergeom([1, -2*n-3, -n], [-3*n/2-1, -3*n/2-1/2], 1/4):
b := n -> (n+3)*binomial(3*n+2, n)/(2*n+3): # A262394(n-1)
a := n -> b(n)*simplify(h(n)):
seq(a(n), n=0..21); # Peter Luschny, Nov 06 2016

Mathematica

Table[(n + 3)*Sum[Binomial[3*n - 2*k + 2, n - k]/(2*n - k + 3), {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Jun 06 2017 *)

Prog

(Maxima)
F(x):=x*(2/sqrt(3*x))*sin((1/3)*asin(sqrt(27*x/4)));
taylor(diff(F(x), x)*F(x)^2/(1-F(x))/x^2, x, 0, 10);
(PARI) for(n=0, 25, print1((n+3)*sum(k=0, n, binomial(3*n-2*k+2, n-k)/(2*n-k+3)), ", ")) \\ G. C. Greubel, Jun 06 2017

Crossrefs

Cf. A001764, A242798, A262394, A277956.

Sequence in context: A104498 A045379 A053487 * A183161 A351151 A263134

Adjacent sequences: A277954 A277955 A277956 * A277958 A277959 A277960

Keyword

nonn

Author

Vladimir Kruchinin, Nov 05 2016

Status

approved