A081185 8th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

0, 1, 16, 194, 2112, 21764, 217280, 2127112, 20562432, 197117968, 1879016704, 17842953248, 168988216320, 1597548359744, 15083504344064, 142288071200896, 1341431869882368, 12641049503662336, 119088016125890560

Offset: 0

Author

Paul Barry, Mar 11 2003

Keywords

• nonn
• easy

Comments

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (16,-62).

Crossrefs

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), this sequence (m=7), A164600 (m=8).

Binomial transform of A081184.

Cf. A081185, A147959.

Programs

Magma

[n le 2 select n-1 else 16*Self(n-1)-62*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 07 2013

Maple

m:=30; S:=series( x/(1-16*x+62*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 12 2021

Mathematica

Join[{a=0,b=1},Table[c=16*b-62*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011 *)
CoefficientList[Series[x/(1-16x+62x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{16,-62},{0,1},30] (* Harvey P. Dale, Sep 24 2013 *)

Sage

[( x/(1-16*x+62*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

Formula

a(n) = 16*a(n-1) - 62*a(n-2), a(0)=0, a(1)=1.

G.f.: x/(1 - 16*x + 62*x^2).

a(n) = ((8 + sqrt(2))^n - (8 - 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(8*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017

From G. C. Greubel, Mar 12 2021: (Start)

a(n) = 2*A147959(n) + 8*A081185(n).

a(n) = (1/2)*Sum_{k=0..n-1} binomial(n-1,k)*7^(n-k-1)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)