A106361 Number of rooted ordered trees where no branch is identical to its adjacent neighbor.
1, 1, 1, 3, 6, 16, 45, 128, 371, 1106, 3343, 10230, 31635, 98714, 310366, 982437, 3128051, 10011848, 32193840, 103955571, 336946034, 1095873115, 3575319049, 11697938232, 38374479841, 126190075741, 415889689954, 1373506798548
Offset: 1
Links
Formula
Shifts left under Carlitz transform.
Carlitz transform T(A(x)) has g.f. 1/(1-sum(k>0, (-1)^(k+1)*A(x^k))).
From Petros Hadjicostas, Sep 17 2017: (Start)
The following results are simple consequences of the fact that the sequence shifts left under the Carlitz transform.
a(n) = Sum_{1 <= s <= n-1} a(n-s)*b(s) for n>=2, where b(n) = Sum_{m|n} (-1)^{1+(n/m)} a(m), with a(1) = 1.
If A(x) = Sum_{n>=1} a(n)*x^n, then A(x)-x = A(x)*Sum_{n>=1} a(n)*x^n/(1+x^n).
(End)