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A217360
a(n) = 2^n*binomial(4*n, n)/(3*n+1).
3
1, 2, 16, 176, 2240, 31008, 453376, 6888960, 107707392, 1721477120, 28000141312, 461964898304, 7712495058944, 130050777006080, 2211737871974400, 37892693797109760, 653389823437701120, 11330548232319664128, 197475886172892823552
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
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
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: A367384 A138014 A206988 * A371669 A363311 A052606
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