%I #25 Apr 16 2022 03:14:03
%S 1,3,8,18,35,62,103,161,241,348,487,664,886,1159,1491,1890,2364,2922,
%T 3574,4329,5198,6192,7322,8600,10039,11651,13450,15450,17665,20110,
%U 22801,25753,28983,32508,36345,40512,45028,49911,55181,60858,66962,73514,80536
%N Number of rooted trees with n nodes and 3 leaves.
%H Georg Fischer, <a href="/A055278/b055278.txt">Table of n, a(n) for n = 4..200</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</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^4*(x^3+x^2+1)/((1-x^2)*(1-x^3)*(1-x)^3) (conjectured). - _Ralf Stephan_, Mar 07 2004
%F a(n) = A055364(4 - n) for all n in Z. - _Michael Somos_, Jun 29 2015
%e G.f. = x^4 + 3*x^5 + 8*x^6 + 18*x^7 + 35*x^8 + 62*x^9 + 103*x^10 + ...
%t a[ n_] := Quotient[ 3 n^4 - 20 n^3 + 54 n^2 - 60 n + 32, 144];
%o (PARI) {a(n) = (3*n^4 - 20*n^3 + 54*n^2 - 60*n + 32) \ 144}; /* _Michael Somos_, Jun 29 2015 */
%o (PARI) {a(n) = if( n<0, n = -1-n; polcoeff( (1 + x + x^3) / ((1 - x)^3 * (1 - x^2) * (1 - x^3)) + x * O(x^n), n), polcoeff( x^4 * (1 + x^2 + 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 A055277.
%Y Cf. A055364.
%K nonn,easy
%O 4,2
%A _Christian G. Bower_, May 09 2000