OFFSET
0,3
COMMENTS
Also appears in the context of pattern avoiding ternary trees.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..400
Nathan Gabriel, Katherine Peske, Lara Pudwell, and Samuel Tay, Pattern Avoidance in Ternary Trees, arXiv:1110.2225 [math.CO], 2011.
N. Gabriel, K. Peske, L. Pudwell, S. Tay, Pattern Avoidance in Ternary Trees, J. Int. Seq. 15 (2012) # 12.1.5.
FORMULA
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
Recurrence (of order 5): 3*n*(3*n + 1)*(3*n + 2)*(10043*n^7 - 178112*n^6 + 1298660*n^5 - 5030058*n^4 + 11133801*n^3 - 14012878*n^2 + 9223088*n - 2422800)*a(n) = 4*(461978*n^10 - 8424141*n^9 + 63797184*n^8 - 260675899*n^7 + 624106521*n^6 - 887032870*n^5 + 715697401*n^4 - 279089746*n^3 + 16511676*n^2 + 18478296*n - 3395520)*a(n-1) - 6*(100430*n^10 - 1981980*n^9 + 16460873*n^8 - 75082304*n^7 + 205450204*n^6 - 344448324*n^5 + 342957257*n^4 - 179367856*n^3 + 27140036*n^2 + 12461424*n - 3772800)*a(n-2) + 2*(n-3)*(100430*n^9 - 1831335*n^8 + 13708728*n^7 - 54564202*n^6 + 124631598*n^5 - 163342351*n^4 + 113612104*n^3 - 30832732*n^2 - 4157040*n + 2954880)*a(n-3) + 2*(n-4)*(n-3)*(20086*n^8 - 316052*n^7 + 2019787*n^6 - 6686515*n^5 + 12045071*n^4 - 11191005*n^3 + 4204160*n^2 + 311748*n - 412560)*a(n-4) - 2*(n-5)*(n-4)*(2*n - 9)*(10043*n^7 - 107811*n^6 + 440891*n^5 - 856933*n^4 + 789434*n^3 - 266000*n^2 - 26568*n + 21744)*a(n-5). - Vaclav Kotesovec, Jan 19 2026
MAPLE
n:=30:
L:=1 - a - x*a + 2*x*a^2 - x^2*a^3 + x^2*a^4:
L:=subs(a=add(q[k]*x^k, k=0..n), L):
Y:=expand(L):
for i from 0 to degree(Y, x) do
p[i]:=coeff(Y, x, i):
od:
S:=solve({ seq(p[t]=0, t=0..n)}, {seq(q[t], t=0..n)}):
normal(subs(S, [seq(q[t], t=0..n)]));
# Alternative:
a:= n-> coeff(series(RootOf(A=1-x*A+2*x*A^2-x^2*A^3+x^2*A^4, A)
, x, n+1), x, n):
seq(a(n), n=0..40); # Alois P. Heinz, Nov 09 2013
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Lara Pudwell, Nov 21 2011
STATUS
approved