A343842 Series expansion of 1/sqrt(8*x^2 + 1), even powers only.

1, -4, 24, -160, 1120, -8064, 59136, -439296, 3294720, -24893440, 189190144, -1444724736, 11076222976, -85201715200, 657270374400, -5082890895360, 39392404439040, -305870434467840, 2378992268083200, -18531097667174400, 144542561803960320, -1128808577897594880

Offset: 0

Author

Peter Luschny, May 04 2021

Keywords

• sign
• easy

Comments

Essentially the inverse binomial convolution of the Delannoy numbers.

Crossrefs

Signed version of A059304.

Cf. A008288, A006139.

Programs

Maple

gf := 1/sqrt(8*x^2 + 1): ser := series(gf, x, 32):
seq(coeff(ser, x, 2*n), n = 0..21);

Mathematica

Take[CoefficientList[Series[1/Sqrt[8*x^2 + 1], {x, 0, 42}], x], {1, -1, 2}] (* Amiram Eldar, May 05 2021 *)

PARI

my(x='x+O('x^25)); Vec(1/sqrt(8*x + 1)) \\ Michel Marcus, May 04 2021

Formula

a(n) = n! * [x^n] BesselJ(0, sqrt(8)*x).

D-finite with recurrence a(n) = 4*(1 - 2*n)*a(n - 1) / n for n >= 2.

a(n) = A(2*n) where A(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A008288(n, k).