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

Author

Gary W. Adamson, Jul 06 2015

Keywords

• nonn
• eigen

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.

Examples

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

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.

Crossrefs

Cf. A086581, A000108, A171199.

Programs

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 *)

Formula

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

Extensions

More terms from Alois P. Heinz, Jul 07 2015