OFFSET
0,2
COMMENTS
8th binomial transform is expansion of 1/sqrt(1-8*x+64*x^2).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
FORMULA
E.g.f.: exp(-4*x)*Bessel_I(0, 2*sqrt(-12)*x).
a(n) = 2^n*Sum_{k=0..n} C(n,n-k)*C(n,k)*(-3)^k.
a(n) = 2^n*A116091(n).
D-finite with recurrence: n*a(n) +4*(2*n-1)*a(n-1) +64*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 07 2012
MATHEMATICA
CoefficientList[Series[1/Sqrt[1+8*x+64*x^2], {x, 0, 30}], x] (* G. C. Greubel, May 10 2019 *)
PROG
(PARI) my(x='x+O('x^30)); Vec(1/sqrt(1+8*x+64*x^2)) \\ G. C. Greubel, May 10 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1+8*x+64*x^2) )); // G. C. Greubel, May 10 2019
(Sage) (1/sqrt(1+8*x+64*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 10 2019
(GAP) List([0..30], n-> 2^n*Sum([0..n], k-> (-3)^k*Binomial(n, k)* Binomial(n, n-k))) # G. C. Greubel, May 10 2019
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Feb 04 2006
STATUS
approved