A259845 a(0)=1, a(1)=3, and the INVERT transform of the sequence deletes the 3.

1, 3, 4, 11, 38, 136, 512, 1993, 7958, 32420, 134216, 563030, 2388092, 10224320, 44127328, 191783029, 838623654, 3686965308, 16287624440, 72262899994, 321852273332, 1438540956048, 6450223722816, 29006443606746, 130790584554748, 591191800834696

Offset

0,2

Comments

The sequence is N = 3 in an infinite set, with the first few being:

A086581, N = 0: (1, 0, 1, 2, 5, 13, 35, 97, ...)

A000108, N = 1: (1, 1, 2, 5, 14, 42, 132, ...)

A171199, N = 2: (1, 2, 3, 8, 25, 83, 289, ...)

... The INVERT transforms of the sequences delete the second terms in the sequences.

The g.f. was contributed by Paul D. Hanna: From the definition of the INVERT transform, 1/(1 - x*A) = A - (N-1)*x. Thus, (1 + (N-1)*x - (1 + (N-1)*x^2)*A) + x*A^2 = 0. The g.f. follows, below.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 16.

Formula

G.f.: A(x) = 1/(2*x) + x - sqrt(1 - 4*x - 4*x^2 + 4*x^4)/(2*x).

Example

The INVERT transform of (1, 3, 4, 11, 38, 136, ...) is (1, 4, 11, 38, 136, ...).

Mathematica

CoefficientList[Series[1/(2*x) + x - Sqrt[1 - 4*x - 4*x^2 + 4*x^4]/(2*x), {x, 0, 25}], x] (* Michael De Vlieger, Jun 12 2024 *)

Crossrefs

Cf. A086581, A000108, A171199.

Sequence in context: A119042 A042273 A179167 * A037185 A299047 A296256

Adjacent sequences: A259842 A259843 A259844 * A259846 A259847 A259848

Keyword

nonn,eigen

Author

Gary W. Adamson, Jul 06 2015

Extensions

More terms from Alois P. Heinz, Jul 07 2015

Status

approved