A374722 Number of nonisomorphic spanning trees of the nC_5-snake with constant distance between cutpoints.

1, 6, 24, 120, 570, 2850, 14100, 70500, 351750, 1758750, 8790000, 43950000, 219731250, 1098656250, 5493187500, 27465937500, 137329218750, 686646093750, 3433228125000, 17166140625000, 85830691406250, 429153457031250, 2145767226562500, 10728836132812500, 53644180371093750

Offset

1,2

Comments

a(n) is the number of spanning trees of the cyclic snake formed with n copies of the cycle on 5 vertices. A cyclic snake is a connected graph whose block-cutpoint is a path and all its n blocks are isomorphic to the cycle C_m.

References

Christian Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60 (2001), 85-96.

Links

Table of n, a(n) for n=1..25.

Index entries for linear recurrences with constant coefficients, signature (5,5,-25).

Formula

a(n) = (3/2)*(3* 5^(n - 2) + 5^floor((n - 2)/2)) for n > 1.

From Stefano Spezia, Jul 23 2024: (Start)

G.f.: x*(1 + x - 11*x^2 - 5*x^3)/((1 - 5*x)*(1 - 5*x^2)).

E.g.f.: (24 + 10*x - 9*cosh(5*x) - 15*cosh(sqrt(5)*x) - 9*sinh(5*x) - 3*sqrt(5)*sinh(sqrt(5)*x))/50. (End)

Example

For n=2, a(2)=6 because there are 6 spanning trees of 2C_5-snake
__ __ __ __ __ __ __ __, __ __ __ __|__ __ __, __ __ __ \/__ __ __,
            __              __           __
__ __ __ __|__ __, __ __ __|__ __, __ __|__ __
                           |            |__

Mathematica

Drop[CoefficientList[Series[x*(1 + x - 11*x^2 - 5*x^3)/((1 - 5*x)*(1 - 5*x^2)), {x, 0, 30}], x], 1] (* Georg Fischer, Aug 09 2024 *)

Prog

(PARI) for(n=1, 30, print1(if(n==1, 1, (3/2)*(3* 5^(n - 2) + 5^floor((n - 2)/2))), ", ")) \\ Georg Fischer, Aug 09 2024

Crossrefs

Cf. A374721.

Sequence in context: A293121 A026982 A051197 * A334328 A324064 A050212

Adjacent sequences: A374719 A374720 A374721 * A374723 A374724 A374725

Keyword

nonn,easy,changed

Author

Christian Barrientos, Jul 17 2024

Extensions

a(25) corrected by Georg Fischer, Aug 09 2024

Status

approved