OFFSET
0,2
COMMENTS
Hankel transform is the (1,-1) Somos-4 sequence A178079.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = A025254(n+2).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(n-k,j)*C(j,k)*C(j+1,k)*2^(n-k-j)/(k+1).
From Vaclav Kotesovec, Mar 02 2014: (Start)
Recurrence: (n+3)*a(n) = 3*(2*n+3)*a(n-1) - 7*n*a(n-2) - (2*n-3)*a(n-3) - (n-3)*a(n-4).
G.f.: (1 - 3*x - x^2 - sqrt(x^4 + 2*x^3 + 7*x^2 - 6*x + 1))/(2*x^3).
a(n) ~ (130-216*r-64*r^2-29*r^3) * sqrt(2*r^3+14*r^2-18*r+4) / (4 * sqrt(Pi) * n^(3/2) * r^n), where r = 1/6*(-3 + sqrt(3*(-11 + (1009 - 24*sqrt(183))^(1/3) + (1009 + 24*sqrt(183))^(1/3))) - sqrt(-66 - 3*(1009 - 24*sqrt(183))^(1/3) - 3*(1009 + 24*sqrt(183))^(1/3) + 216*sqrt(3/(-11 + (1009 - 24*sqrt(183))^(1/3) + (1009 + 24*sqrt(183))^(1/3))))) = 0.23742047190096998... is the root of the equation r^4 + 2*r^3 + 7*r^2 - 6*r + 1 = 0.
(End)
MATHEMATICA
Table[Sum[Sum[Binomial[n-k, j]*Binomial[j, k]*Binomial[j+1, k]*2^(n-k-j)/(k+1), {j, 0, n-k}], {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 02 2014 *)
CoefficientList[Series[(1-3*x-x^2 -Sqrt[x^4+2*x^3+7*x^2-6*x+1])/(2*x^3), {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *)
PROG
(PARI) a(n)=sum(k=0, floor(n/2), sum(j=0, n-k, binomial(n-k, j)*binomial(j, k)*binomial(j+1, k)*2^(n-k-j)/(k+1)));
vector(22, n, a(n-1))
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 -3*x-x^2 - Sqrt(x^4+2*x^3+7*x^2-6*x+1))/(2*x^3))); // G. C. Greubel, Aug 14 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul Barry, Dec 26 2010
STATUS
approved