A003762 Number of spanning trees with degrees 1 and 3 in D_4 X P_n.

1, 4, 16, 92, 432, 1884, 8582, 39736, 181936, 829672, 3793850, 17366388, 79441576, 363298928, 1661695126, 7601017276, 34767611570, 159026305464, 727389859704, 3327116203688, 15218354613018, 69609219627912

Offset

1,2

References

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.

F. Faase, Counting Hamiltonian cycles in product graphs

F. Faase, Results from the counting program

Index entries for sequences related to trees

Index entries for linear recurrences with constant coefficients, signature (4,-5,30,13,36,48,-76,-14,-36,4,8,-4).

Formula

Faase gives a 12-term linear recurrence on his web page:

a(1) = 1,

a(2) = 4,

a(3) = 16,

a(4) = 92,

a(5) = 432,

a(6) = 1884,

a(7) = 8582,

a(8) = 39736,

a(9) = 181936,

a(10) = 829672,

a(11) = 3793850,

a(12) = 17366388,

a(13) = 79441576,

a(14) = 363298928,

a(15) = 1661695126 and

a(n) = 4a(n-1) - 5a(n-2) + 30a(n-3) + 13a(n-4) + 36a(n-5) + 48a(n-6) - 76a(n-7) - 14a(n-8) - 36a(n-9) + 4a(n-10) + 8a(n-11) - 4a(n-12).

G.f.: -x*(8*x^14 +16*x^13 +34*x^12 +12*x^11 +8*x^10 +4*x^9 -4*x^8 +20*x^7 -46*x^6 -48*x^5 -11*x^4 -18*x^3 -5*x^2 -1)/(4*x^12 -8*x^11 -4*x^10 +36*x^9 +14*x^8 +76*x^7 -48*x^6 -36*x^5 -13*x^4 -30*x^3 +5*x^2 -4*x +1). - Colin Barker, Sep 07 2012

Mathematica

CoefficientList[Series[-(8 * x^14 + 16 * x^13 + 34 * x^12 + 12 * x^11 + 8 * x^10 + 4 * x^9 - 4 * x^8 + 20 * x^7 - 46 * x^6 - 48 * x^5 - 11 * x^4 - 18 * x^3 - 5 * x^2 - 1)/(4 * x^12 - 8 * x^11 - 4 * x^10 + 36 * x^9 + 14 * x^8 + 76 * x^7 - 48 * x^6 - 36 * x^5 - 13 * x^4 - 30 * x^3 + 5 * x^2 - 4 * x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 14 2013 *)

Crossrefs

Sequence in context: A331648 A009568 A139155 * A143501 A111291 A050913

Adjacent sequences: A003759 A003760 A003761 * A003763 A003764 A003765

Keyword

nonn,easy

Author

Frans J. Faase

Extensions

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

Status

approved