A055330 Number of rooted identity trees with n nodes and 5 leaves.

3, 26, 116, 387, 1068, 2587, 5678, 11540, 22034, 39957, 69366, 116009, 187823, 295574, 453582, 680625, 1000952, 1445516, 2053343, 2873165, 3965216, 5403347, 7277330, 9695538, 12787847, 16708973, 21642067, 27802808, 35443793, 44859494, 56391551, 70434706

Offset: 10

Author

Christian G. Bower, May 12 2000

Keywords

• nonn
• easy

Links

• Andrew Howroyd, Table of n, a(n) for n = 10..1000
• Index entries for sequences related to rooted trees
• Index entries for linear recurrences with constant coefficients, signature (4,-4,-3,7,-3,0,1,0,-3,0,3,0,-1,0,3,-7,3,4,-4,1).

Crossrefs

Column 5 of A055327.

Cf. A061347, A257145.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( x^10*(3 + 14*x+24*x^2+36*x^3+41*x^4+38*x^5+29*x^6+16*x^7+6*x^8+3*x^9)/((1-x)^3*(&*[1-x^j: j in [1..5]])) )); // G. C. Greubel, Nov 09 2023

Mathematica

Drop[CoefficientList[Series[x^10*(3+14*x+24*x^2+36*x^3+41*x^4+38*x^5+29*x^6 +16*x^7+6*x^8+3*x^9)/((1-x)^3*Product[1-x^j, {j,5}]), {x,0,40}], x], 10] (* G. C. Greubel, Nov 09 2023 *)

SageMath

def p(x): return 3 +14*x +24*x^2 +36*x^3 +41*x^4 +38*x^5 +29*x^6 +16*x^7 +6*x^8 +3*x^9
def A055330_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^10*p(x)/((1-x)^3*product(1-x^j for j in range(1,6))) ).list()
a=A055330_list(50); a[10:] # G. C. Greubel, Nov 09 2023

Formula

G.f.: x^10*(3 +14*x +24*x^2 +36*x^3 +41*x^4 +38*x^5 +29*x^6 +16*x^7 +6*x^8 +3*x^9)/((1-x)^9*(1+x)^3*(1+x^2)*(1+x+x^2)*(1+x+x^2+x^3+x^4)). - Colin Barker, Nov 07 2012

a(n) = (1/(8*10!))*(5303207 -25330590*n [phone]*n^2 -18286800*n^3 +7777980*n^4 -1990044*n^5 +286440*n^6 -21240*n^7 +630*n^8) -(-1)^n*(89 - 34*n +4*n^2)/2048 -(3/64)*(-1)^binomial(n+1,2) -A061347(n+1)/81 +A257145(n+2)/25. - G. C. Greubel, Nov 09 2023