A055329 - OEIS

%I #20 Nov 10 2023 10:55:09

%S 2,11,40,109,254,524,998,1774,2995,4833,7525,11346,16659,23877,33528,

%T 46203,62637,83643,110213,143432,184600,235129,296687,371072,460382,

%U 566866,693121,841917,1016422,1220001,1456473,1729878,2044767,2405940

%N Number of rooted identity trees with n nodes and 4 leaves.

%H Andrew Howroyd, <a href="/A055329/b055329.txt">Table of n, a(n) for n = 8..1000</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-4,3,-1,3,0,-3,1,-3,4,1,-3,1).

%F G.f.: x^8*(2+5*x+9*x^2+8*x^3+5*x^4+x^6)/((1-x)^7*(1+x)^3*(1+x^2)*(1+x+x^2)) - _Colin Barker_, Nov 07 2012

%F a(n) = -[n=0] + (60*n^6 -1236*n^5 +9450*n^4 -33520*n^3 +59940*n^2 -66294*n +48065)/69120 +(-1)^n*(4*n^2 -38*n +89)/512 +(3/32)*(-1)^floor((n+1)/2) + ChebyshevU(n, -1/2)/27. - _G. C. Greubel_, Nov 09 2023

%t Drop[CoefficientList[Series[x^8*(2+5*x+9*x^2+8*x^3+5*x^4+x^6)/((1-x)^3*(1-x^2)^2*(1-x^3)*(1-x^4)), {x,0,50}], x], 8] (* _G. C. Greubel_, Nov 09 2023 *)

%o (PARI) Vec((2 + 5*x + 9*x^2 + 8*x^3 + 5*x^4 + x^6)/((1 - x)^7*(1 + x)^3*(1 + x^2)*(1 + x + x^2)) + O(x^40)) \\ _Andrew Howroyd_, Aug 28 2018

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( x^8*(2+5*x+9*x^2+8*x^3+5*x^4+x^6)/((1-x)^3*(1-x^2)^2*(1-x^3)*(1-x^4)) )); // _G. C. Greubel_, Nov 09 2023

%o (SageMath)

%o def A055329_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( x^8*(2+5*x+9*x^2+8*x^3+5*x^4+x^6)/((1-x)^3*(1-x^2)^2*(1-x^3)*(1-x^4)) ).list()

%o a=A055329_list(50); a[8:] # _G. C. Greubel_, Nov 09 2023

%Y Column 4 of A055327.

%K nonn,easy

%O 8,1

%A _Christian G. Bower_, May 12 2000