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Expansion of 1/((1-x)*(1-2*x)^4).
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%I #49 Aug 24 2022 09:24:50

%S 1,9,49,209,769,2561,7937,23297,65537,178177,471041,1216513,3080193,

%T 7667713,18808833,45547521,109051905,258473985,607125505,1414529025,

%U 3271557121,7516192769,17163091969

%N Expansion of 1/((1-x)*(1-2*x)^4).

%H Michael De Vlieger, <a href="/A027608/b027608.txt">Table of n, a(n) for n = 0..3288</a>

%H Michał Adamaszek and Henry Adams, <a href="https://arxiv.org/abs/2103.01040">On Vietoris-Rips complexes of hypercube graphs</a>, arXiv:2103.01040 [math.CO], 2021.

%H M. H. Albert, M. D. Atkinson, and R. Brignall, <a href="https://doi.org/10.37236/2442">The enumeration of three pattern classes using monotone grid classes</a>, E. J. Combinat. 19 (3) (2012) P20. Chapter 5.5.1 (with leading zeros).

%H Harry Crane, <a href="https://ajc.maths.uq.edu.au/pdf/61/ajc_v61_p057.pdf">Left-right arrangements, set partitions, and pattern avoidance</a>, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.

%H Santiago López de Medrano, <a href="https://arxiv.org/abs/2003.07508">On the genera of moment-angle manifolds associated to dual-neighborly polytopes, combinatorial formulas and sequences</a>, arXiv:2003.07508 [math.GT], 2020.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (9,-32,56,-48,16).

%F a(n-1) = 1 + (n-1)*2^(n+1) + ((n^3 - 7*n + 6)*2^(n-1))/3, n >= 1. - Roger Voles, Dec 07 2004, index corrected by _R. J. Mathar_, Mar 14 2011

%F a(n) = A119258(n+4,n). - _Reinhard Zumkeller_, May 11 2006

%F a(n) = 1 + n*2^(n+2) + (((n+1)^3 - 7*(n+1) + 6)*2^n)/3 = (n/3)*(n^2 + 3*n + 8)*2^n + 1, n >= 0. - _Daniel Forgues_, Nov 01 2012

%F E.g.f.: exp(x) + (8/3)*x*(3 + 3*x + x^2)*exp(2*x). - _G. C. Greubel_, Aug 24 2022

%t CoefficientList[Series[1/((1-x)*(1-2x)^4), {x, 0, 22}], x] (* _Michael De Vlieger_, Jun 23 2020 *)

%t LinearRecurrence[{9,-32,56,-48,16},{1,9,49,209,769},30] (* _Harvey P. Dale_, Apr 09 2021 *)

%o (PARI) Vec(1/((1-x)*(1-2*x)^4)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012

%o (Magma) [(n/3)*(n^2+3*n+8)*2^n +1: n in [0..40]]; // _G. C. Greubel_, Aug 24 2022

%o (SageMath) [(n/3)*(n^2+3*n+8)*2^n + 1 for n in (0..40)] # _G. C. Greubel_, Aug 24 2022

%Y Cf. A001789 (first differences).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_