A081184 7th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

0, 1, 14, 149, 1428, 12989, 114730, 995737, 8548008, 72872473, 618458246, 5233409213, 44200191420, 372832446869, 3142245259426, 26468308629121, 222870793614672, 1876180605036721, 15791601170624510, 132901927952017253

Offset: 0

Author

Paul Barry, Mar 11 2003

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• 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 (14,-47).

Crossrefs

Binomial transform of A081183.

Cf. A081185.

Programs

Magma

[n le 2 select n-1 else 14*Self(n-1)-47*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 07 2013

Mathematica

CoefficientList[Series[x/(1-14*x+47*x^2), {x,0,30}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{14,-47},{0,1},30] (* Harvey P. Dale, Nov 12 2013 *)

SageMath

A081184=BinaryRecurrenceSequence(14,-47,0,1)
[A081184(n) for n in range(31)] # G. C. Greubel, Jan 14 2024

Formula

a(n) = 14*a(n-1) - 47*a(n-2), a(0)=0, a(1)=1.

G.f.: x/(1 - 14*x + 47*x^2). [Corrected by Georg Fischer, May 15 2019]

a(n) = ((7 + sqrt(2))^n - (7 - sqrt(2))^n)/(2*sqrt(2)).

a(n) = Sum_{k=0..n} C(n,2*k+1) * 2^k * 7^(n-2*k-1).

E.g.f.: exp(7*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017