A081180 4th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

0, 1, 8, 50, 288, 1604, 8800, 47944, 260352, 1411600, 7647872, 41420576, 224294400, 1214467136, 6575615488, 35602384000, 192760455168, 1043650265344, 5650555750400, 30593342288384, 165638957801472, 896804870374400

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.

Index entries for linear recurrences with constant coefficients, signature (8,-14).

Formula

a(n) = 8a(n-1) - 14a(n-2), a(0)=0, a(1)=1.

G.f.: x/(1 - 8x + 14x^2).

a(n) = ((4 + sqrt(2))^n - (4 - sqrt(2))^n/(2*sqrt(2)).

a(n) = Sum_{k=0..n} C(n,2k+1) 2^k*4^(n-2k-1).

If shifted once left, fourth binomial transform of A143095. - Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009, R. J. Mathar, Oct 15 2009

E.g.f.: exp(4*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017

Mathematica

Join[{a=0, b=1}, Table[c=8*b-14*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
CoefficientList[Series[x / (1 - 8 x + 14 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{8, -14}, {0, 1}, 30] (* Harvey P. Dale, Aug 17 2019 *)

Prog

(Sage) [lucas_number1(n, 8, 14) for n in range(0, 22)] # Zerinvary Lajos, Apr 23 2009
(Magma) I:=[0, 1]; [n le 2 select I[n] else 8*Self(n-1)-14*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 06 2013

Crossrefs

Binomial transform of A081179.

Cf. A081182.

Sequence in context: A133357 A081675 A283277 * A052177 A127745 A243876

Adjacent sequences: A081177 A081178 A081179 * A081181 A081182 A081183

Keyword

easy,nonn

Author

Paul Barry, Mar 11 2003

Extensions

Modified the completing comment on the fourth binomial transform - R. J. Mathar, Oct 15 2009

Status

approved