A116093 Expansion of 1/sqrt(1+4*x+12*x^2).

1, -2, 0, 16, -56, 48, 384, -1920, 3168, 8512, -66560, 161280, 113920, -2224640, 7311360, -3354624, -69253632, 306754560, -408059904, -1898029056, 12054196224, -25377005568, -38874316800, 443400781824, -1289598418944, -52751204352, 15086928789504, -58620595404800

Offset

0,2

Comments

Apart from signs identical this is to A098336. - Joerg Arndt, Jun 30 2013

Fourth binomial transform of the expansion of 1/sqrt(1-4*x+12*x^2), A098336.

Links

Seiichi Manyama, 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(-2*x)*Bessel_I(0,2*sqrt(-2)*x).

a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*binomial(n-k,k)(-2)^(n-k).

D-finite with recurrence: n*a(n) +2*(2*n-1)*a(n-1) +12*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 07 2012

G.f.: G(0), where G(k)= 1 - 2*x*(1+3*x)*(4*k+1)/( 2*k+1 - x*(1+3*x)*(2*k+1)*(4*k+3)/(x*(1+3*x)*(4*k+3) - (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jun 30 2013

Mathematica

CoefficientList[Series[1/Sqrt[1+4x+12x^2], {x, 0, 30}], x] (* Harvey P. Dale, Oct 15 2014 *)

Prog

(PARI) my(x='x+O('x^30)); Vec(1/sqrt(1+4*x+12*x^2)) \\ G. C. Greubel, May 10 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1+4*x+12*x^2) )); // G. C. Greubel, May 10 2019
(Sage) (1/sqrt(1+4*x+12*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 10 2019

Crossrefs

Column 2 of A307819.

Cf. A098336.

Sequence in context: A338448 A012414 A098336 * A012409 A012661 A345652

Adjacent sequences: A116090 A116091 A116092 * A116094 A116095 A116096

Keyword

easy,sign

Author

Paul Barry, Feb 04 2006

Status

approved