A319436 Number of palindromic plane trees with n nodes.

1, 1, 2, 3, 6, 10, 20, 35, 68, 122, 234, 426, 808, 1484, 2798, 5167, 9700, 17974, 33656, 62498, 116826, 217236, 405646, 754938, 1408736, 2623188, 4892848, 9114036, 16995110, 31664136, 59034488, 110004243, 205068892, 382156686, 712363344, 1327600346, 2474618434

Offset: 1

Author

Gus Wiseman, Sep 18 2018

Keywords

• nonn

Comments

A rooted plane tree is palindromic if the sequence of branches directly under any given node is a palindrome.

Examples

			The a(7) = 20 palindromic plane trees:
  ((((((o))))))  (((((oo)))))  ((((ooo))))  (((oooo)))  ((ooooo))  (oooooo)
                 ((((o)(o))))  (((o(o)o)))  ((o(oo)o))  (o(ooo)o)
                 (((o))((o)))  ((o((o))o))  (o((oo))o)  (oo(o)oo)
                               (((o)o(o)))  ((oo)(oo))
                               (o(((o)))o)  ((o)oo(o))
                               ((o)(o)(o))  (o(o)(o)o)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500
• Gus Wiseman, The a(8) = 35 palindromic plane trees.
• Gus Wiseman, The a(11) = 234 palindromic plane trees.

Crossrefs

Cf. A000108, A000670, A001003, A005043, A008965, A025065, A032128, A118376, A242414, A317085, A317086, A317087, A319122, A319437.

Programs

Mathematica

panplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[panplane/@c],#==Reverse[#]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[panplane[n]],{n,10}]

PARI

PAL(p)={(1+p)/subst(1-p, x, x^2)}
seq(n)={my(p=O(1));for(i=1, n, p=PAL(x*p)); Vec(p)} \\ Andrew Howroyd, Sep 19 2018

Formula

a(n) ~ c * d^n, where d = 1.86383559155190653688720443906758855085492625375... and c = 0.24457511051198663873739022949952908293770055... - Vaclav Kotesovec, Nov 16 2021