This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A217364 #62 Sep 08 2022 08:46:04
%S A217364 1,2,20,280,4560,80960,1520064,29680640,596593920,12262581760,
%T A217364 256556410880,5445566730240,116974976102400,2538140268625920,
%U A217364 55548588652625920,1224777962595287040,27180522769369333760,606648758810237337600
%N A217364 a(n) = 2^n*binomial(5*n, n)/(4*n+1).
%C A217364 Old name was: Series reversion of x - 2*x^5.
%C A217364 Regular blocks of 3 intermediate zeros have been left out of the sequence: If y = x - 2x^5, then x = y + 2*y^5 + 20*y^9 + 280*y^13 + 4560*y^17 - ...
%C A217364 a(n) is the number of lattice paths (Schroeder paths) from (0,0) to (n,5n) with unit steps N=(0,1), E=(1,0) and D=(1,1) staying weakly above the line y = 5x with the total number of occurrences of NE and D equal to n. - _Michael D. Weiner_, Aug 21 2019
%H A217364 Alois P. Heinz, <a href="/A217364/b217364.txt">Table of n, a(n) for n = 0..724</a> (terms n=1..76 from Robert Israel)
%H A217364 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.
%H A217364 R. J. Mathar, <a href="http://arxiv.org/abs/1211.3963">Series Expansion of Generalized Fresnel Integrals</a>, arXiv:1211.3963 [math.CA], 2012.
%F A217364 D-finite with recurrence (4*n+1)*(4*n)*(4*n-1)*(4*n-2)*a(n) - 10*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1) = 0.
%F A217364 a(n) ~ 5^(5*n)/(2^(7*n+4)*sqrt(Pi/10)*n^(3/2)). - _Vaclav Kotesovec_, Aug 20 2013
%F A217364 a(n) = Gamma(n+1/5)*Gamma(n+2/5)*Gamma(n+3/5)*Gamma(n+4/5)*6250^(n+1)*sqrt(5)/ (25000*Pi^2*Gamma(4*n+2)). - _Robert Israel_, May 19 2014
%F A217364 a(n) = 5*2^n*Gamma(5*n)/(Gamma(n)*Gamma(4*n+2)) for n > 0. - _Peter Luschny_, May 19 2014
%F A217364 G.f.: F([1/5, 2/5, 3/5, 4/5], [1/2, 3/4, 5/4], 3125*x/128), where F is the generalized hypergeometric function. - _Stefano Spezia_, Sep 03 2019
%F A217364 G.f. A(x) satisfies: A(x) = 1 / (1 - 2 * x * A(x)^4). - _Ilya Gutkovskiy_, Nov 12 2021
%p A217364 A217364 := j -> pochhammer(1/5, j)*pochhammer(2/5, j)*pochhammer(3/5, j)*pochhammer(4/5, j)*6250^j/(4*j+1)!:
%p A217364 seq(A217364(n),n=0..100); # _Robert Israel_, May 19 2014
%t A217364 Table[CoefficientList[InverseSeries[Series[x-2*x^5,{x,0,100}],x],x][[4*n-2]],{n,1,20}] (* _Vaclav Kotesovec_,Aug 20 2013 *)
%t A217364 Table[2^n Binomial[5 n, n] / (4 n + 1), {n, 0, 20}] (* _Michael D. Weiner_, Aug 21 2019 *)
%o A217364 (Sage)
%o A217364 A217364 = lambda n: 5*2^n*gamma(5*n)/(gamma(n)*gamma(4*n+2)) if n > 0 else 1
%o A217364 [A217364(n) for n in (0..18)] # _Peter Luschny_, May 19 2014
%o A217364 (PARI) for(n=0,20, print1(round(if(n==0, 1, 5*2^n*gamma(5*n)/(gamma(n)*gamma(4*n+2)))), ", ")) \\ _G. C. Greubel_, Apr 01 2017
%o A217364 (PARI) {my(x='x+O('x^133), v=Vec(serreverse(x - 2*x^5))); vector(#v\4, n, v[4*n-3] ) } \\ _Joerg Arndt_, Apr 02 2017
%o A217364 (Magma) [2^n*Binomial(5*n, n)/(4*n+1): n in [0..25]]; // _Michael D. Weiner_, Aug 21 2019
%Y A217364 Cf. A153231 (x-2*x^3), A217360 (x-2*x^4).
%K A217364 nonn,easy
%O A217364 0,2
%A A217364 _R. J. Mathar_, Oct 01 2012
%E A217364 New definition and offset from _Michael D. Weiner_, Sep 03 2019