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). (Formerly M1243)

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

0,4

Comments

This is a rescaled version of the number of spanning trees in the cube of an n-cycle. See A331905 for details. - N. J. A. Sloane, Feb 06 2020

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).

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]

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 *)

Prog

(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
(Magma) m:=40; R<x>:=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

Crossrefs

Cf. A169630, A331905.

Sequence in context: A278346 A277867 A278595 * A331653 A286293 A167801

Adjacent sequences: A005819 A005820 A005821 * A005823 A005824 A005825

Keyword

nonn,easy

Author

N. J. A. Sloane

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.

Status

approved