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Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

In the following guide to p-INVERT sequences using s = (1,1,0,0,0,...) = A019590, in some cases t(1,1,0,0,0,...) is a shifted version of the cited sequence:

p(S) t(1,1,0,0,0,...)

1 - S A000045 (Fibonacci numbers)

1 - S^2 A094686

1 - S^3 A115055

1 - S^4 A291379

1 - S^5 A281380

1 - S^6 A281381

1 - 2 S A002605

1 - 3 S A125145

(1 - S)^2 A001629

(1 - S)^3 A001628

(1 - S)^4 A001629

(1 - S)^5 A001873

(1 - S)^6 A001874

1 - S - S^2 A123392

1 - 2 S - S^2 A291382

1 - S - 2 S^2 A124861

1 - 2 S - S^2 A291383

(1 - 2 S)^2 A073388

(1 - 3 S)^2 A291387

(1 - 5 S)^2 A291389

(1 - 6 S)^2 A291391

(1 - S)(1 - 2 S) A291393

(1 - S)(1 - 3 S) A291394

(1 - 2 S)(1 - 3 S) A291395

(1 - S)(1 - 2 S) A291393

(1 - S)(1 - 2 S)(1 - 3 S) A291396

1 - S - S^3 A291397

1 - S^2 - S^3 A291398

1 - S - S^2 - S^3 A186812

1 - S - S^2 - S^3 - S^4 A291399

1 - S^2 - S^4 A291400

1 - S - S^4 A291401

1 - S^3 - S^4 A291402

1 - 2 S^2 - S^4 A291403

1 - S^2 - 2 S^4 A291404

1 - 2 S^2 - 2 S^4 A291405

1 - S^3 - S^6 A291407

(1 - S)(1 - S^2) A291408

(1 - S^2)(1 - S)^2 A291409

1 - S - S^2 - 2 S^3 A291410

1 - 2 S - S^2 + S^3 A291411

1 - S - 2 S^2 + S^3 A291412

1 - 3 S + S^2 + S^3 A291413

1 - 2 S + S^3 A291414

1 - 3 S + S^2 A291415

1 - 4 S + S^2 A291416

1 - 4 S + 2 S^2 A291417

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.)

...B...C.......A...B

....\./.........\./

.A...x....-->....x...C.................A..().........()..A..

..\./.............\./...................\./....-->....\./...

...x...............x.....................x.............x....

(a . (b . c)) -> ((a . b) . c) ____ (a . ()) --> (() . a)

That is, we rotate the binary tree left, in case it is possible and otherwise (if the right hand side of a tree is a terminal node) swap the left and right subtree (so that the terminal node ends to the left hand side), i.e., apply the automorphism *A069770. Look at the example in A069770 to see how this will produce the given sequence of integers.

This is the first multiclause nonrecursive automorphism in table A089840 and the first one whose order is not finite, i.e., the maximum size of cycles in this permutation is not bounded (see A089842). The cycle counts in range [A014137(n-1)..A014138(n)] of this permutation is given by A001683(n+1), which is otherwise the same sequence as for Catalan automorphisms *A057161/*A057162, but shifted once right. For an explanation, please see the notes in OEIS Wiki.

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.)

A...B..............B...C

.\./................\./

..x...C..-->.....A...x................()..B.......B..()

...\./............\./..................\./...-->...\./.

....x..............x....................x...........x..

((a . b) . c) -> (a . (b . c)) __ (() . b) --> (b . ())

That is, we rotate the binary tree right, in case it is possible and otherwise (if the left hand side of a tree is a terminal node) swap the right and left subtree (so that the terminal node ends to the right hand side), i.e. apply the automorphism *A069770. Look at the example in A069770 to see how this will produce the given sequence of integers.

See also the comments at A074679.

Dirichlet inverse of this sequence is A154271. There is progression of sequences starting with A000007, A019590 and then this sequence A154272. From A019590 onwards the Dirichlet inverse of such sequences appears to be positive as often as negative. Except for the first term, the all 1's sequence A000012, is a union of the 1's in sequences A000007, A019590, A154272 etc. It therefore in a sense seems likely that the Moebius function is positive as often as negative because the Dirichlet inverse of A000012 is the Moebius function A008683. However the whole is more than the sum of its parts and the likelihood of the signs of the Moebius function cannot be inferred.

With offset -1, this is the replicable function number 1a. The generating function A(x) = 1/x + x is the first modular "fiction". This is a completely 2-replicable function. The g.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v + 2 - u^2. Similarly the replicable function number 2b gives the sequence 1,0,-1,0,0,0,0,0,... - Michael Somos, Aug 04 2009

The g.f. x+x^3 equals x*Phi(4,x), where Phi are the cyclotomic polynomials. The series reversion of y=x+x^3 is x = y - y^3 + 3*y^5 - 12*y^7 + 55*y^9 - ..., which is a signed variant of A001764. - R. J. Mathar, Sep 29 2012

From Andrey Zabolotskiy, Mar 12 2018: (Start)

If reflections are not allowed, we get A145392. If any rotations and reflections are allowed, we get A054346.

The parent lattice of the sublattices under consideration has Patterson symmetry group p4mm, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145392 (p4), A145394 (p6), A003051 (p6mm).

Rutherford says at p. 161 that a(n) != A054346(n) only when A002654(n) > 2, but actually these two sequence differ at other terms, too, for example, at n = 30 (see illustration). (End)

Also the numerators of the series expansion of log(1+x). Denominators are A028310. - Robert G. Wilson v, Aug 14 2015

Each row of this table gives the counts of elements fixed by the Catalan bijection (given in the corresponding row of A073200) when it acts on A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

Eigensequence of the triangle = A051163: (1, 2, 5, 12, 30, 76,...)

Another version of A152815. - Philippe Deléham, Dec 13 2008

Row sums : A016116(n); Diagonal sums: A000931(n+5). - Philippe Deléham, Dec 13 2008

Triangle, with zeros omitted, given by (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 16 2012

Sums along rising diagonals are A134816. - John Molokach, Jul 09 2013

Multiplicative with a(p^e) = (if p=2 then A019590(e) else 1), p prime and e>0.

Period 8 Repeat: [0, 1, 1, 1, 1, 1, 1, 1]. - Wesley Ivan Hurt, Jun 21 2014

Matrix inverse of A000012.

Moebius transform of the sequence A000035. Dirichlet inverse of A209229. Partial sums of a(n) is characteristic function of 1 (A063524). a(n)=(-1)^(n+1)*A019590(n). a(n) for n >= 1 is Dirichlet convolution of following functions b(n), c(n), a(n) = Sum_{d|n} b(d)*c(n/d): a(n) = A000012(n) * A092673(n). Examples of Dirichlet convolutions with function a(n), i.e. b(n) = Sum_{d|n} a(d)*c(n/d): a(n) * A000012(n) = A000035(n), a(n) * A000027(n) = A026741(n), a(n) * A008683(n) = A092673(n), a(n) * A036987(n-1) = A063524(n), a(n) * A000005(n) = A001227(n). - Jaroslav Krizek, Mar 21 2009

The Kn21 sums, see A180662, of triangle A108299 equal the terms of this sequence. - Johannes W. Meijer, Aug 14 2011

{a(n-1)}A132393.%20-%20_Wolfdieter%20Lang">{n>=1}, gives the alternating row sums of A132393. - _Wolfdieter Lang, May 09 2017

With offset 0 the alternating row sums of A097805. - Peter Luschny, Sep 07 2017