A277956 a(n) = (n+2)*Sum_{i=0..n}(binomial(3*n-2*i+1, n-i)/(2*n-i+2)).

1, 4, 19, 101, 573, 3382, 20483, 126292, 788878, 4976489, 31635811, 202354517, 1300880374, 8398175713, 54409200963, 353571026085, 2303666554659, 15043760670031, 98439176169692, 645290365460761, 4236768489465944, 27857102370774193

Offset

0,2

Links

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

Formula

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

From Vaclav Kotesovec, Nov 06 2016: (Start)

Recurrence: 2*(n+1)*(2*n + 1)*(91*n^4 - 232*n^3 + 15*n^2 + 266*n - 120)*a(n) = (2821*n^6 - 4189*n^5 - 10027*n^4 + 18573*n^3 - 3498*n^2 - 3968*n + 960)*a(n-1) - (2821*n^6 - 4189*n^5 - 10027*n^4 + 18573*n^3 - 3498*n^2 - 3968*n + 960)*a(n-2) + 3*(3*n - 5)*(3*n - 4)*(91*n^4 + 132*n^3 - 135*n^2 - 36*n + 20)*a(n-3).

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

a(n) = A026004(n)*hypergeom([1,-2*n-2,-n],[-3*n/2-1/2,-3*n/2],1/4). - Peter Luschny, Nov 06 2016

Maple

h := n -> hypergeom([1, -2*n-2, -n], [-3*n/2-1/2, -3*n/2], 1/4):
b := n -> binomial(3*n+1, n)*(n+2)/(2*n+2): # A026004
a := n -> `if`(n=0, 1, b(n)*simplify(h(n))):
seq(a(n), n=0..21); # Peter Luschny, Nov 06 2016

Mathematica

f[n_] := (n + 2)Sum[ Binomial[3n - 2i + 1, n - i]/(2n - i + 2), {i, 0, n}]; Array[f, 22, 0] (* Robert G. Wilson v, Nov 06 2016 *)

Prog

(Maxima)
F(x):=x*(2/sqrt(3*x))*sin((1/3)*asin(sqrt(27*x/4)));
taylor(diff(F(x), x)*F(x)/(1-F(x))/x, x, 0, 10);
(PARI) for(n=0, 25, print1((n+2)*sum(i=0, n, (binomial(3*n-2*i+1, n-i)/(2*n-i+2))), ", ")) \\ G. C. Greubel, Apr 09 2017

Crossrefs

Cf. A001764, A026004, A242798.

Sequence in context: A086624 A307121 A261490 * A307678 A151382 A234958

Adjacent sequences: A277953 A277954 A277955 * A277957 A277958 A277959

Keyword

nonn

Author

Vladimir Kruchinin, Nov 05 2016

Status

approved