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%I #22 Jul 25 2024 08:46:38
%S 1,-4,24,-160,1120,-8064,59136,-439296,3294720,-24893440,189190144,
%T -1444724736,11076222976,-85201715200,657270374400,-5082890895360,
%U 39392404439040,-305870434467840,2378992268083200,-18531097667174400,144542561803960320,-1128808577897594880
%N Series expansion of 1/sqrt(8*x^2 + 1), even powers only.
%C Essentially the inverse binomial convolution of the Delannoy numbers.
%F a(n) = n! * [x^n] BesselJ(0, sqrt(8)*x).
%F D-finite with recurrence a(n) = 4*(1 - 2*n)*a(n - 1) / n for n >= 2.
%F a(n) = A(2*n) where A(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A008288(n, k).
%p gf := 1/sqrt(8*x^2 + 1): ser := series(gf, x, 32):
%p seq(coeff(ser, x, 2*n), n = 0..21);
%t Take[CoefficientList[Series[1/Sqrt[8*x^2 + 1], {x, 0, 42}], x], {1, -1, 2}] (* _Amiram Eldar_, May 05 2021 *)
%o (PARI) my(x='x+O('x^25)); Vec(1/sqrt(8*x + 1)) \\ _Michel Marcus_, May 04 2021
%Y Signed version of A059304.
%Y Cf. A008288, A006139.
%K sign,easy
%O 0,2
%A _Peter Luschny_, May 04 2021