A081571 Sixth binomial transform of F(n+1).

1, 7, 50, 363, 2669, 19814, 148153, 1113615, 8402722, 63577171, 481991621, 3659227062, 27808295345, 211479529943, 1609093780114, 12247558413819, 93245414394973, 710040492168070, 5407464407991017, 41185377124992351, 313703861897268866, 2389549742539808867

Offset

0,2

Comments

Binomial transform of A081570.

Case k=6 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 (13,-41).

Formula

a(n) = 13*a(n-1) - 41*a(n-2), a(0)=1, a(1)=7.

a(n) = (1/2 - sqrt(5)/10)*(13/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)*(sqrt(5)/2 + 13/2)^n.

G.f.: (1 - 6*x)/(1 - 13*x + 41*x^2).

a(n) = Sum_{k=0..n} A094441(n,k)*6^k. - Philippe Deléham, Dec 14 2009

E.g.f.: exp(13*x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Mar 30 2023

Maple

seq(coeff(series((1-6*x)/(1-13*x+41*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Aug 12 2019

Mathematica

CoefficientList[Series[(1-6x)/(1 -13x +41x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 09 2013 *)

Prog

(Magma) I:=[1, 7]; [n le 2 select I[n] else 13*Self(n-1)-41*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 09 2013
(PARI) my(x='x+O('x^30)); Vec((1-6*x)/(1-13*x+41*x^2)) \\ G. C. Greubel, Aug 12 2019
(Sage)
def A081571_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P((1-6*x)/(1-13*x+41*x^2)).list()
A081571_list(30) # G. C. Greubel, Aug 12 2019
(GAP) a:=[1, 7];; for n in [3..30] do a[n]:=13*a[n-1]-41*a[n-2]; od; a; # G. C. Greubel, Aug 12 2019

Crossrefs

Cf. A000045, A028387, A081570.

Sequence in context: A022037 A054413 A163458 * A275827 A081189 A108869

Adjacent sequences: A081568 A081569 A081570 * A081572 A081573 A081574

Keyword

easy,nonn

Author

Paul Barry, Mar 22 2003

Status

approved