A271897 Expansion of ( 1-2*x+3*x^2 ) / ( 1-4*x+5*x^2-4*x^3 ).

1, 2, 6, 18, 50, 134, 358, 962, 2594, 6998, 18870, 50866, 137106, 369574, 996230, 2685474, 7239042, 19513718, 52601558, 141793810, 382222322, 1030326470, 2777369510, 7486734978, 20181398242, 54401396118, 146645533174, 395300745074, 1065580898898, 2872402002918, 7742906497478, 20871939570914

Offset: 0

Author

R. J. Mathar, Apr 16 2016

Keywords

• nonn
• easy

Comments

Sum of all second elements at level n of the TRIP-Stern sequence corresponding to the permutation triple (e,e,e)

Number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>4, 3>2} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the last element, and the third element is larger than the second one. - Sergey Kitaev, Dec 08 2020

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Ilya Amburg, Krishna Dasaratha, Laure Flapan, Thomas Garrity, Chansoo Lee, Cornelia Mihaila, Nicholas Neumann-Chun, Sarah Peluse, Matthew Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239 [math.CO], 2015, Section 7.2.1.
• Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
• Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
• Index entries for linear recurrences with constant coefficients, signature (4,-5,4).

Crossrefs

Cf. A000930 (maximum at level n).

Programs

Mathematica

CoefficientList[Series[(1-2x+3x^2)/(1-4x+5x^2-4x^3),{x,0,40}],x] (* or *) LinearRecurrence[{4,-5,4},{1,2,6},40] (* Harvey P. Dale, Dec 27 2016 *)

PARI

x='x+O('x^99); Vec((1-2*x+3*x^2)/(1-4*x+5*x^2-4*x^3)) \\ Altug Alkan, Apr 16 2016