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%I #22 Jan 11 2024 10:07:40
%S 1,1,3,12,54,261,1324,6954,37493,206316,1154050,6542485,37507919,
%T 217081155,1266646114,7443100944,44008522719,261631301144,
%U 1562969609155,9377744249277,56486588669929,341452466500382,2070684006442310,12594325039504367,76808163066135791
%N Generating function satisfies A(x)=1-xA(x)+2x(A(x))^2-x^2(A(x))^3+x^2(A(x))^4.
%C Also appears in the context of pattern avoiding ternary trees.
%H Alois P. Heinz, <a href="/A200740/b200740.txt">Table of n, a(n) for n = 0..400</a>
%H Nathan Gabriel, Katherine Peske, Lara Pudwell, and Samuel Tay, <a href="http://arxiv.org/abs/1110.2225">Pattern Avoidance in Ternary Trees</a>, arXiv:1110.2225 [math.CO], 2011.
%H N. Gabriel, K. Peske, L. Pudwell, S. Tay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Pudwell/pudwell.html">Pattern Avoidance in Ternary Trees</a>, J. Int. Seq. 15 (2012) # 12.1.5.
%F D-finite with recurrence 9*n*(3*n+2)*(9637049385113*n -13150529182719) *(3*n+1)*a(n) +3*(-1867245272511941*n^4 +3938815702450522*n^3 -1829703422934531*n^2 -40908425929938*n +80119802734368)*a(n-1) +3*(1493812832168185*n^4 -11081290962331766*n^3 +30368478809400583*n^2 -37445753408742482*n +17518049080170408)*a(n-2) +2*(-3598515629532857*n^4 +51918859363655934*n^3 -272767116207263419*n^2 +607523860755165342*n -484618766805936168)*a(n-3) +2*(2236067158786314*n^4 -43895348429968415*n^3 +313117370004358791*n^2 -968797762596343960*n +1100409843957627312)*a(n-4) +4*(-320929802901755*n^4 +7262334358284366*n^3 -58466924652690997*n^2 +201695420809801662*n -253987363669614120)*a(n-5) +2*(n-6) *(56449675289272*n^3 -1539119961654835*n^2 +11344967150541329*n -25343984173596980)*a(n-6) +2*(n-6) *(n-7) *(88750351258118*n^2 -884530270439421*n +2059300905886804)*a(n-7) +4*-(6920632454763*n -35058029508284)*(n-7)*(n-8)*(2*n-15)*a(n-8)=0. - _R. J. Mathar_, Jan 11 2024
%p n:=30:
%p L:=1 - a - x*a + 2*x*a^2 - x^2*a^3 + x^2*a^4:
%p L:=subs(a=add(q[k]*x^k,k=0..n),L):
%p Y:=expand(L):
%p for i from 0 to degree(Y,x) do
%p p[i]:=coeff(Y,x,i):
%p od:
%p S:=solve({ seq(p[t]=0, t=0..n)}, {seq(q[t], t=0..n)}):
%p normal(subs(S,[seq(q[t], t=0..n)]));
%p # second Maple program:
%p a:= n-> coeff(series(RootOf(A=1-x*A+2*x*A^2-x^2*A^3+x^2*A^4, A)
%p , x, n+1), x, n):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Nov 09 2013
%t A[_] = 0; Do[A[x_] = 1 - x A[x] + 2x A[x]^2 - x^2 A[x]^3 + x^2 A[x]^4 + O[x]^25, {25}]; CoefficientList[A[x], x] (* _Jean-François Alcover_, Nov 28 2018 *)
%K nonn
%O 0,3
%A _Lara Pudwell_, Nov 21 2011
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