A081037 Inverse binary transform of A027656.
1, -1, 3, -7, 16, -36, 80, -176, 384, -832, 1792, -3840, 8192, -17408, 36864, -77824, 163840, -344064, 720896, -1507328, 3145728, -6553600, 13631488, -28311552, 58720256, -121634816, 251658240, -520093696, 1073741824, -2214592512, 4563402752, -9395240960, 19327352832
Offset: 0
Links
- Vincenzo Librandi and Evert Provoost, Table of n, a(n) for n = 0..1000 [Terms 0 through 300 were computed by Vincenzo Librandi; terms 301 through 1000 by Evert Provoost, Jan 25 2016]
- F. Ellermann, Illustration of binomial transforms
- Index entries for linear recurrences with constant coefficients, signature (-4,-4).
Programs
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Magma
I:=[1, -1, 3, -7]; [n le 4 select I[n] else -4*Self(n-1)-4*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 06 2013
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Mathematica
CoefficientList[Series[(1 + x)^3 / (1 + 2 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *) LinearRecurrence[{-4,-4},{1,-1,3,-7},40] (* Harvey P. Dale, Apr 14 2019 *) -
PARI
Vec((1+x)^3/(1+2*x)^2 + O(x^40)) \\ Michel Marcus, Jan 25 2016
Formula
G.f.: (1+x)^3/(1+2*x)^2.
a(n) = (-1)^n*A045623(n+1)/4, n>1. - R. J. Mathar, Sep 27 2012
Recurrence: a(n) = -4a(n-1) - 4a(n-2), a(0)=1, a(1)=-1, a(2)=3, a(3)=-7. - Ralf Stephan, Jul 14 2013
E.g.f.: exp(-x)*(2*cosh(x) + x*sinh(x))/2. - Stefano Spezia, Apr 24 2023