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A217360 a(n) = 2^n*binomial(4*n, n)/(3*n+1). 2
1, 2, 16, 176, 2240, 31008, 453376, 6888960, 107707392, 1721477120, 28000141312, 461964898304, 7712495058944, 130050777006080, 2211737871974400, 37892693797109760, 653389823437701120, 11330548232319664128, 197475886172892823552 (list; graph; refs; listen; history; text; internal format)
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, M. D. Weiner, Schröder Coloring and Applications, arXiv:1908.08103 [math.CO], 2019.

FORMULA

Apparently 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

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).

Sequence in context: A108999 A138014 A206988 * A052606 A011553 A291816

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

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Last modified May 5 19:09 EDT 2021. Contains 343573 sequences. (Running on oeis4.)