A200740 Generating function satisfies A(x)=1-xA(x)+2x(A(x))^2-x^2(A(x))^3+x^2(A(x))^4.

1, 1, 3, 12, 54, 261, 1324, 6954, 37493, 206316, 1154050, 6542485, 37507919, 217081155, 1266646114, 7443100944, 44008522719, 261631301144, 1562969609155, 9377744249277, 56486588669929, 341452466500382, 2070684006442310, 12594325039504367, 76808163066135791

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

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)]));
# second Maple program:
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

Sequence in context: A158826 A107264 A370441 * A177133 A186241 A193115

Adjacent sequences: A200737 A200738 A200739 * A200741 A200742 A200743

Keyword

nonn

Author

Lara Pudwell, Nov 21 2011

Status

approved