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A055328 Number of rooted identity trees with n nodes and 3 leaves.

%I A055328 #31 Nov 22 2023 01:44:01
%S A055328 1,5,13,28,53,91,146,223,326,461,634,851,1119,1446,1839,2307,2859,
%T A055328 3504,4252,5114,6100,7222,8492,9922,11525,13315,15305,17510,19945,
%U A055328 22625,25566,28785,32298,36123,40278,44781,49651,54908,60571,66661
%N A055328 Number of rooted identity trees with n nodes and 3 leaves.
%H A055328 Andrew Howroyd, <a href="/A055328/b055328.txt">Table of n, a(n) for n = 6..1000</a>
%H A055328 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A055328 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,0,1,2,-3,1).
%F A055328 G.f.: x^6*(1+2*x)/((1-x^2)*(1-x^3)*(1-x)^3).
%F A055328 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-5) + 2*a(n-6) - 3*a(n-7) + a(n-8) for n>13. - _Colin Barker_, Sep 06 2019
%F A055328 a(n) = (1/288)*(41 - 240*n + 216*n^2 - 64*n^3 + 6*n^4 - 9*(-1)^n - 32*ChebyshevU(n, -1/2)). - _G. C. Greubel_, Nov 09 2023
%e A055328 Illustration for a(7)=5 from _N. J. A. Sloane_, Mar 21 2016:
%e A055328 The five 7-node rooted identity trees with 3 leaves are:
%e A055328 (O denotes the root)
%e A055328 o
%e A055328 |
%e A055328 o o o
%e A055328 |/ /
%e A055328 o o
%e A055328 |/
%e A055328 O
%e A055328 ..........
%e A055328 o
%e A055328 |
%e A055328 o   o
%e A055328 |  /
%e A055328 o o o
%e A055328 |//
%e A055328 O
%e A055328 ..........
%e A055328 o
%e A055328 |
%e A055328 o
%e A055328 |
%e A055328 o o
%e A055328 |/
%e A055328 o o
%e A055328 |/
%e A055328 O
%e A055328 ..............
%e A055328 o
%e A055328 |
%e A055328 o o
%e A055328 |/
%e A055328 o
%e A055328 |
%e A055328 o o
%e A055328 |/
%e A055328 O
%e A055328 ..............
%e A055328 o
%e A055328 |
%e A055328 o o
%e A055328 |/
%e A055328 o o
%e A055328 |/
%e A055328 o
%e A055328 |
%e A055328 O
%e A055328 ..............
%t A055328 LinearRecurrence[{3,-2,-1,0,1,2,-3,1}, {1,5,13,28,53,91,146,223}, 40] (* _Jean-François Alcover_, Sep 06 2019 *)
%o A055328 (PARI) Vec((2*x+1)/((1-x^2)*(1-x^3)*(1-x)^3) + O(x^40)) \\ _Andrew Howroyd_, Aug 28 2018
%o A055328 (Magma) [(9*(1-(-1)^n) -272*n +216*n^2 -64*n^3 +6*n^4 +96*Floor((n+2)/3))/288: n in [6..46]]; // _G. C. Greubel_, Nov 09 2023
%o A055328 (SageMath) [(9*(n%2) -136*n +108*n^2 -32*n^3 +3*n^4 +48*((n+2)//3))/144 for n in range(6,47)] # _G. C. Greubel_, Nov 09 2023
%Y A055328 Column 3 of A055327.
%K A055328 nonn,easy
%O A055328 6,2
%A A055328 _Christian G. Bower_, May 12 2000