OFFSET
0,3
COMMENTS
Functional equation for the g.f. is derived from the recurrence of the pendular triangle A122445. Iterated convolutions of this sequence with the central terms (A122447) generates all diagonals of A122445. For example: A122448 = A122446 * A122447; A122449 = A122446^2 * A122447.
Diagonal sums of triangle T with T(n,k) = 2^k*A133336(n,k). - Philippe Deléham, Nov 10 2009
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: A(x) = (1 + 2*x^2 - sqrt(1 -4*x -4*x^2 +4*x^4))/(2*x*(1+2*x)).
Recurrence: (n+1)*a(n) = 2*(n-2)*a(n-1) + 12*(n-1)*a(n-2) + 8*(n-2)*a(n-3) - 4*(n-5)*a(n-4) - 8*(n-5)*a(n-5). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = (1/6)*(6+sqrt(6*(10 + 2^(2/3)*(43-3*sqrt(177))^(1/3) + 2^(2/3)*(43+3*sqrt(177))^(1/3))) + sqrt(6*(20-2^(2/3)*(43-3*sqrt(177))^(1/3) - 2^(2/3)*(43+3*sqrt(177))^(1/3) + 24*sqrt(6/(10+2^(2/3)*(43-3*sqrt(177))^(1/3) + 2^(2/3)*(43+3*sqrt(177))^(1/3)))))) = 4.797536514160165558... is the root of the equation 4 - 4*d^2 - 4*d^3 + d^4 = 0 and c = 0.908214882020417619380249683... is the positive root of the equation -59 - 944*c^2 - 2032*c^4 - 320*c^6 + 5184*c^8 = 0. - Vaclav Kotesovec, Sep 17 2013, updated Mar 18 2024
MAPLE
m:=30; S:=series( (1+2*x^2 -sqrt(1-4*x-4*x^2+4*x^4))/(2*x*(1+2*x)), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 16 2021
MATHEMATICA
CoefficientList[Series[(1+2*x^2-Sqrt[1-4*x-4*x^2+4*x^4])/(2*x*(1+2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 17 2013 *)
PROG
(PARI) {a(n)=polcoeff(2/(1+2*x^2+sqrt(1-4*x-4*x^2+4*x^4+x*O(x^n))), n)}
(Sage)
def A122446_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( (1+2*x^2 -sqrt(1-4*x-4*x^2+4*x^4))/(2*x*(1+2*x)) ).list()
A122446_list(30) # G. C. Greubel, Mar 16 2021
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1+2*x^2 -Sqrt(1-4*x-4*x^2+4*x^4))/(2*x*(1+2*x)) )); // G. C. Greubel, Mar 16 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 07 2006
STATUS
approved