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A081185
8th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).
12
0, 1, 16, 194, 2112, 21764, 217280, 2127112, 20562432, 197117968, 1879016704, 17842953248, 168988216320, 1597548359744, 15083504344064, 142288071200896, 1341431869882368, 12641049503662336, 119088016125890560
OFFSET
0,3
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
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)
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 *)
PROG
(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
(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
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.
Sequence in context: A081202 A196803 A292785 * A159517 A153599 A016280
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Mar 11 2003
STATUS
approved