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

0,5

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.

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

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[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 *)

Crossrefs

Cf. A001696, A039941, A268107.

Sequence in context: A035055 A119559 A329675 * 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