A027608 Expansion of 1/((1-x)*(1-2*x)^4).
1, 9, 49, 209, 769, 2561, 7937, 23297, 65537, 178177, 471041, 1216513, 3080193, 7667713, 18808833, 45547521, 109051905, 258473985, 607125505, 1414529025, 3271557121, 7516192769, 17163091969
Offset: 0
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..3288
- Michał Adamaszek and Henry Adams, On Vietoris-Rips complexes of hypercube graphs, arXiv:2103.01040 [math.CO], 2021.
- M. H. Albert, M. D. Atkinson, and R. Brignall, The enumeration of three pattern classes using monotone grid classes, E. J. Combinat. 19 (3) (2012) P20. Chapter 5.5.1 (with leading zeros).
- Harry Crane, Left-right arrangements, set partitions, and pattern avoidance, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.
- Santiago López de Medrano, On the genera of moment-angle manifolds associated to dual-neighborly polytopes, combinatorial formulas and sequences, arXiv:2003.07508 [math.GT], 2020.
- Index entries for linear recurrences with constant coefficients, signature (9,-32,56,-48,16).
Crossrefs
Cf. A001789 (first differences).
Programs
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Magma
[(n/3)*(n^2+3*n+8)*2^n +1: n in [0..40]]; // G. C. Greubel, Aug 24 2022
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Mathematica
CoefficientList[Series[1/((1-x)*(1-2x)^4), {x, 0, 22}], x] (* Michael De Vlieger, Jun 23 2020 *) LinearRecurrence[{9,-32,56,-48,16},{1,9,49,209,769},30] (* Harvey P. Dale, Apr 09 2021 *) -
PARI
Vec(1/((1-x)*(1-2*x)^4)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012
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SageMath
[(n/3)*(n^2+3*n+8)*2^n + 1 for n in (0..40)] # G. C. Greubel, Aug 24 2022
Formula
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
a(n) = A119258(n+4,n). - Reinhard Zumkeller, May 11 2006
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
E.g.f.: exp(x) + (8/3)*x*(3 + 3*x + x^2)*exp(2*x). - G. C. Greubel, Aug 24 2022