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Number of asymmetric mobiles (circular rooted trees) with n nodes and 3 leaves.
3

%I #19 Mar 16 2022 05:54:53

%S 1,4,10,22,42,73,119,184,272,389,540,731,969,1261,1614,2037,2538,3126,

%T 3811,4603,5512,6550,7728,9058,10553,12226,14090,16160,18450,20975,

%U 23751,26794,30120,33747,37692,41973,46609,51619,57022,62839,69090,75796,82979

%N Number of asymmetric mobiles (circular rooted trees) with n nodes and 3 leaves.

%H Georg Fischer, <a href="/A055364/b055364.txt">Table of n, a(n) for n = 5..129</a>

%H <a href="/index/Mo#mobiles">Index entries for sequences related to mobiles</a>

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

%F a(n) = A055278(4 - n) for all n in Z. - _Michael Somos_, Jun 29 2015

%e G.f. = x^5 + 4*x^6 + 10*x^7 + 22*x^8 + 42*x^9 + 73*x^10 + 119*x^11 + ...

%t a[ n_] := Quotient[ 3 n^4 - 28 n^3 + 102 n^2 - 180 n + 144, 144];

%o (PARI) {a(n) = (3*n^4 - 28*n^3 + 102*n^2 - 180*n + 144) \ 144}; /* _Michael Somos_, Jun 29 2015 */

%o (PARI) {a(n) = if( n<1, n = -n; polcoeff( (1 + x^2 + x^3) / ((1 - x)^3 * (1 - x^2) * (1 - x^3)) + x * O(x^n), n), polcoeff( x^5 * (1 + x + x^3) / ((1 - x)^3 * (1 - x^2) * (1 - x^3)) + x * O(x^n), n))}; /* _Michael Somos_, Jun 29 2015 */

%Y Column 3 of A055363.

%Y Cf. A055278.

%K nonn

%O 5,2

%A _Christian G. Bower_, May 16 2000