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

%I A217360 #51 Sep 08 2022 08:46:04
%S A217360 1,2,16,176,2240,31008,453376,6888960,107707392,1721477120,
%T A217360 28000141312,461964898304,7712495058944,130050777006080,
%U A217360 2211737871974400,37892693797109760,653389823437701120,11330548232319664128,197475886172892823552
%N A217360 a(n) = 2^n*binomial(4*n, n)/(3*n+1).
%C A217360 Old name was: Series reversion of x - 2*x^4.
%C A217360 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 + ...
%C A217360 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
%H A217360 Jinyuan Wang, <a href="/A217360/b217360.txt">Table of n, a(n) for n = 0..100</a>
%H A217360 D. Birmajer, J. B. Gil, J. D. Gil and M. D. Weiner, <a href="http://arxiv.org/abs/1908.08103">Schröder Coloring and Applications</a>, arXiv:1908.08103 [math.CO], 2019.
%F A217360 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).
%F A217360 a(n) = [x^(3*n)](f(x)/x) where f(x) is the reversion of x - 2*x^4.
%F A217360 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
%F A217360 G.f. A(x) satisfies: A(x) = 1 / (1 - 2 * x * A(x)^3). - _Ilya Gutkovskiy_, Nov 12 2021
%p A217360 A100089 := proc(n)
%p A217360     (3*n+1)! ;
%p A217360 end proc:
%p A217360 A060706 := proc(n)
%p A217360     (4*n)!/n!/4^n ;
%p A217360 end proc:
%p A217360 A217360 := proc(n)
%p A217360     8^(n)*A060706(n)/A100089(n) ;
%p A217360 end proc:
%p A217360 seq(A217360(n), n=0..20);
%t A217360 Table[2^n Binomial[4 n, n] / (3 n + 1), {n, 0, 20}] (* _Vincenzo Librandi_, Jul 26 2019 *)
%o A217360 (Magma) [2^n*Binomial(4*n, n)/(3*n+1): n in [0..25]]; // _Vincenzo Librandi_, Jul 26 2019
%Y A217360 Cf. A153231 (x+2*x^3).
%K A217360 nonn,easy
%O A217360 0,2
%A A217360 _R. J. Mathar_, Oct 01 2012
%E A217360 Offset decreased by 1 and name changed by _Michael D. Weiner_, Jul 25 2019