A005822 G.f.: x*(1-x^2)*(x^4+x^3-x^2+x+1) / (x^8-4*x^6-x^4-4*x^2+1).
0, 1, 1, 2, 4, 11, 16, 49, 72, 214, 319, 947, 1408, 4187, 6223, 18502, 27504, 81769, 121552, 361379, 537196, 1597106, 2374129, 7058377, 10492416, 31194361, 46371025, 137862866, 204935836, 609282227, 905709904, 2692710841, 4002767136, 11900382694, 17690150767
Offset: 0
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Colin Barker, Table of n, a(n) for n = 0..1000 [Jul 09 2015; a(0) inserted by _Georg Fischer_, Jan 27 2020]
- G. Baron et al., The number of spanning trees in the square of a cycle, Fib. Quart., 23 (1985), 258-264.
- Tsuyoshi Miezaki, A note on spanning trees.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- Index entries for sequences related to trees
- Index entries for linear recurrences with constant coefficients, signature (0,4,0,1,0,4,0,-1).
Programs
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Magma
m:=40; R
:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!( x*(1-x^2)*(x^4+x^3-x^2+x+1) / (x^8-4*x^6-x^4-4*x^2+1))); // Vincenzo Librandi, Jan 28 2020 -
Maple
A005822:=-z*(z-1)*(1+z)*(z**4+z**3-z**2+z+1)/(-4*z**6-z**4-4*z**2+1+z**8); # [Conjectured (correctly) by Simon Plouffe in his 1992 dissertation; adapted to offset 0 by Georg Fischer, Jan 27 2020]
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Mathematica
CoefficientList[Series[x (1 - x^2) (x^4 + x^3 - x^2 + x + 1) / (x^8 - 4 x^6 - x^4 - 4 x^2 + 1), {x, 0, 35}], x] (* Vincenzo Librandi, Jan 28 2020 *) -
PARI
Vec(-x*(x-1)*(x+1)*(x^4+x^3-x^2+x+1)/(x^8-4*x^6-x^4-4*x^2+1) + O(x^50)) \\ Colin Barker, Jul 09 2015
Extensions
G.f. adapted to the offset from Colin Barker, Jul 09 2015
Entry revised by N. J. A. Sloane, Jan 25 2020 and Feb 06 2020.
Comments