A045761 Define polynomials Pn by P0 = 0, P1 = x, P2 = P1 + P0, P3 = P2 * P1, P4 = P3 + P2, etc. alternately adding or multiplying. For even n > 2k, then first k coefficients of Pn remain unchanged and their values are the first k terms of the sequence.
0, 1, 1, 1, 2, 3, 6, 12, 24, 50, 107, 232, 508, 1124, 2513, 5665, 12858, 29356, 67371, 155345, 359733, 836261, 1950829, 4565305, 10714501, 25212843, 59474318, 140609809, 333126672, 790764280, 1880489541, 4479494059, 10687448937, 25536624382, 61102431113
Offset: 0
Keywords
Examples
The sequence of polynomials is 0, x, x, x^2, x^2 + x, x^4 + x^3, x^4 + x^3 + x^2 + x, ..., and after this all the even polynomials end with x^3 + x^2 + x (+ 0), so the first 4 terms of the sequence are these coefficients (in ascending order): 0, 1, 1, 1. - _Michael B. Porter_, Aug 09 2016
Links
- Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..2000 (terms 0..1000 from Alois P. Heinz)
- Jean-Luc Baril, Sergey Kirgizov, Armen Petrossian, Dyck paths with a first return decomposition constrained by height, Submitted, 2017.
Programs
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Mathematica
k = 32; P[0] = 0; P[1] = x; P[n_] := P[n] = If[EvenQ[n], P[n-1] + P[n-2], P[n-1]*P[n-2]] + O[x]^(2k+1) // Normal; CoefficientList[P[2k], x][[1 ;; k+1]] (* Jean-François Alcover, Aug 07 2016 *)
Formula
a(n) ~ c * d^n / n^(3/2), where d = 2.50297436517909273228379630... and c = 0.34042564735836570861482... . - Vaclav Kotesovec, Aug 08 2016, updated Aug 27 2016
Conjecture: 1/d = 0.39952466709679946... = A268107. - Jean-François Alcover, Aug 08 2016
Extensions
More terms from Michael Somos, May 19 2000