A217360 a(n) = 2^n*binomial(4*n, n)/(3*n+1).

1, 2, 16, 176, 2240, 31008, 453376, 6888960, 107707392, 1721477120, 28000141312, 461964898304, 7712495058944, 130050777006080, 2211737871974400, 37892693797109760, 653389823437701120, 11330548232319664128, 197475886172892823552, 3457235990102304358400, 60771273100594637701120

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

Daniel Birmajer, Juan B. Gil, Juan D. Gil, and Michael D. Weiner, Schröder Coloring and Applications, Journal of Integer Sequences, Vol. 24 (2021), Article 21.1.3; arXiv preprint, 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

a(n) ~ 2^(9*n+1/2) / (3^(3*n+3/2) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 17 2025

Maple

A100089 := proc(n)
    (3*n+1)! ;
end proc:
A060706 := proc(n)
    (4*n)!/n!/4^n ;
end proc:
A217360 := proc(n)
    8^(n)*A060706(n)/A100089(n) ;
end proc:
seq(A217360(n), n=0..20);

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

Cf. A153231 (x+2*x^3), A002293, A060706, A100089.

Sequence in context: A367384 A138014 A206988 * A381819 A371669 A363311

Adjacent sequences: A217357 A217358 A217359 * A217361 A217362 A217363

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