A081570 Fifth binomial transform of F(n+1).

1, 6, 37, 233, 1490, 9633, 62753, 410926, 2700349, 17786985, 117346714, 774991289, 5121849473, 33865596822, 223987930325, 1481764925737, 9803764203682, 64870223394129, 429263295428641, 2840659771285310, 18798621916707821

Offset: 0

Author

Paul Barry, Mar 22 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A081569.

Case k=5 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2), a(0)=1, a(1)=k+1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (11,-29).

Crossrefs

Cf. A000045.

Programs

GAP

a:=[1,6];; for n in [3..30] do a[n]:=11*a[n-1]-29*a[n-2]; od; a; # G. C. Greubel, Aug 12 2019

Magma

I:=[1, 6]; [n le 2 select I[n] else 11*Self(n-1)-29*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 09 2013

Maple

seq(coeff(series((1-5*x)/(1-11*x+29*x^2), x, n+1), x, n), n = 0 .. 30); # G. C. Greubel, Aug 12 2019

Mathematica

CoefficientList[Series[(1-5x)/(1 -11x +29x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 09 2013 *)
LinearRecurrence[{11,-29},{1,6},30] (* Harvey P. Dale, Aug 04 2022 *)

PARI

my(x='x+O('x^30)); Vec((1-5*x)/(1-11*x+29*x^2)) \\ G. C. Greubel, Aug 12 2019

Sage

def A081570_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-5*x)/(1-11*x+29*x^2)).list()
A081570_list(30) # G. C. Greubel, Aug 12 2019

Formula

a(n) = 11*a(n-1) - 29*a(n-2), a(0)=1, a(1)=6.

a(n) = (1/2 - sqrt(5)/10)*(11/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)*(sqrt(5)/2 + 11/2)^n .

G.f.: (1-5*x)/(1-11*x+29*x^2).

a(n) = Sum_{k=0..n} A094441(n,k)*5^k. - Philippe Deléham, Dec 14 2009