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A374722
Number of nonisomorphic spanning trees of the nC_5-snake with constant distance between cutpoints.
1
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.
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
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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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
a(25) corrected by Georg Fischer, Aug 09 2024
STATUS
approved