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A004306 Rook polynomials.
(Formerly M1670)
4
1, 1, 2, 6, 24, 44, 80, 144, 264, 484, 888, 1632, 3000, 5516, 10144, 18656, 34312, 63108, 116072, 213488, 392664, 722220, 1328368, 2443248, 4493832, 8265444, 15202520, 27961792, 51429752, 94594060, 173985600, 320009408, 588589064 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is the number of perfect matchings in the circulant graph with 2*n vertices with jumps 1 and 3. - Robert Israel, Jan 24 2019

REFERENCES

Lehmer, D. H.; Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..400

N. Metropolis, M. L. Stein, P. R. Stein, Permanents of cyclic (0,1) matrices, Journal of Combinatorial Theory, Volume 7, Issue 4, December 1969, Pages 291-321.

Earl Glen Whitehead, Jr., Four-discordant permutations, J. Austral. Math. Soc. Ser. A 28 (1979), no. 3, 369-377.

Index entries for linear recurrences with constant coefficients, signature (2, 0, 0, -1).

FORMULA

G.f.: (1-x+2*x^3+13*x^4-3*x^5-6*x^6-10*x^7)/(1-2*x+x^4).

a(0)=1, a(1)=1, a(2)=2, a(3)=6, a(4)=24, a(5)=44, a(6)=80, a(7)=144, a(n)=2*a(n-1)-a(n-4). - Harvey P. Dale, Dec 13 2011

MATHEMATICA

Join[{1, 1, 2, 6}, LinearRecurrence[{2, 0, 0, -1}, {24, 44, 80, 144}, 40]] (* or *) CoefficientList[Series[(1-x+2x^3+13x^4- 3x^5- 6x^6- 10x^7)/ (1-2x+ x^4), {x, 0, 40}], x] (* Harvey P. Dale, Dec 13 2011 *)

PROG

(PARI) my(x='x+O('x^40)); Vec((1-x+2*x^3+13*x^4-3*x^5-6*x^6-10*x^7)/(1 -2*x+x^4)) \\ G. C. Greubel, Apr 22 2019

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+ 2*x^3+13*x^4-3*x^5-6*x^6-10*x^7)/(1-2*x+x^4) )); // G. C. Greubel, Apr 22 2019

(Sage) ((1-x+2*x^3+13*x^4-3*x^5-6*x^6-10*x^7)/(1-2*x+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019

CROSSREFS

Cf. A000803. 4th column of A008305.

Equals 2 * (A001644(n) + 1), n>3.

Sequence in context: A090755 A192196 A000496 * A092485 A113904 A099144

Adjacent sequences:  A004303 A004304 A004305 * A004307 A004308 A004309

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 27 01:52 EST 2020. Contains 332299 sequences. (Running on oeis4.)