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
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).
Programs
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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
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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 *) -
Sage
[lucas_number1(n,8,14) for n in range(0, 22)] # Zerinvary Lajos, Apr 23 2009
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
Extensions
Modified the completing comment on the fourth binomial transform - R. J. Mathar, Oct 15 2009