OFFSET
0,2
COMMENTS
Old name was: Series reversion of x - 2*x^4.
Regular blocks of 2 intermediate zeros have been removed from the sequence: If y = x - 2*x^4, then x = y + 2*y^4 + 16*y^7 + 176*y^10 + 2240*y^13 + 31008*y^16 + ...
a(n) is the number of lattice paths (Schroeder paths) from (0,0) to (n,4n) with unit steps N=(0,1), E=(1,0) and D=(1,1) staying weakly above the line y = 4x with the total number of occurrences of NE and D equal to n. - Michael D. Weiner, Jul 25 2019
LINKS
Jinyuan Wang, Table of n, a(n) for n = 0..100
D. Birmajer, J. B. Gil, J. D. Gil and M. D. Weiner, Schröder Coloring and Applications, arXiv:1908.08103 [math.CO], 2019.
FORMULA
D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n)- 8*(4*n-1)*(4*n-3)*(4*n-2)*a(n-1) = 0, so a(n) = 8^n*A060706(n)/A100089(n) = 2^n*A002293(n).
a(n) = [x^(3*n)](f(x)/x) where f(x) is the reversion of x - 2*x^4.
G.f.: F([1/4, 1/2, 3/4], [2/3, 4/3], 512*x/27), where F is the generalized hypergeometric function. - Stefano Spezia, Aug 18 2019
G.f. A(x) satisfies: A(x) = 1 / (1 - 2 * x * A(x)^3). - Ilya Gutkovskiy, Nov 12 2021
MAPLE
MATHEMATICA
Table[2^n Binomial[4 n, n] / (3 n + 1), {n, 0, 20}] (* Vincenzo Librandi, Jul 26 2019 *)
PROG
(Magma) [2^n*Binomial(4*n, n)/(3*n+1): n in [0..25]]; // Vincenzo Librandi, Jul 26 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, Oct 01 2012
EXTENSIONS
Offset decreased by 1 and name changed by Michael D. Weiner, Jul 25 2019
STATUS
approved