

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.


2



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
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OFFSET

0,5


LINKS

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..2000 (terms 0..1000 from Alois P. Heinz)
JeanLuc Baril, Sergey Kirgizov, Armen Petrossian, Dyck paths with a first return decomposition constrained by height, Submitted, 2017.


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.  JeanFrançois Alcover, Aug 08 2016


EXAMPLE

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


MATHEMATICA

k = 32; P[0] = 0; P[1] = x;
P[n_] := P[n] = If[EvenQ[n], P[n1] + P[n2], P[n1]*P[n2]] + O[x]^(2k+1) // Normal;
CoefficientList[P[2k], x][[1 ;; k+1]] (* JeanFrançois Alcover, Aug 07 2016 *)


CROSSREFS

Cf. A001696, A039941, A268107.
Sequence in context: A042950 A035055 A119559 * A187741 A216632 A077903
Adjacent sequences: A045758 A045759 A045760 * A045762 A045763 A045764


KEYWORD

nonn


AUTHOR

James Boudinot (jboudinot(AT)yahoo.com)


EXTENSIONS

More terms from Michael Somos, May 19 2000


STATUS

approved



