A217364 a(n) = 2^n*binomial(5*n, n)/(4*n+1).

1, 2, 20, 280, 4560, 80960, 1520064, 29680640, 596593920, 12262581760, 256556410880, 5445566730240, 116974976102400, 2538140268625920, 55548588652625920, 1224777962595287040, 27180522769369333760, 606648758810237337600

Offset: 0

Author

R. J. Mathar, Oct 01 2012

Keywords

• nonn
• easy

Comments

Old name was: Series reversion of x - 2*x^5.

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 - ...

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..724 (terms n=1..76 from Robert Israel)
• D. Birmajer, J. B. Gil, J. D. Gil and M. D. Weiner, Schröder Coloring and Applications, arXiv:1908.08103 [math.CO], 2019.
• R. J. Mathar, Series Expansion of Generalized Fresnel Integrals, arXiv:1211.3963 [math.CA], 2012.

Crossrefs

Cf. A153231 (x-2*x^3), A217360 (x-2*x^4).

Programs

Magma

[2^n*Binomial(5*n, n)/(4*n+1): n in [0..25]]; // Michael D. Weiner, Aug 21 2019

Maple

A217364 := j -> pochhammer(1/5, j)*pochhammer(2/5, j)*pochhammer(3/5, j)*pochhammer(4/5, j)*6250^j/(4*j+1)!:
seq(A217364(n),n=0..100); # Robert Israel, May 19 2014

Mathematica

Table[CoefficientList[InverseSeries[Series[x-2*x^5,{x,0,100}],x],x][[4*n-2]],{n,1,20}] (* Vaclav Kotesovec,Aug 20 2013 *)
Table[2^n Binomial[5 n, n] / (4 n + 1), {n, 0, 20}] (* Michael D. Weiner, Aug 21 2019 *)

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

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

Sage

A217364 = lambda n: 5*2^n*gamma(5*n)/(gamma(n)*gamma(4*n+2)) if n > 0 else 1
[A217364(n) for n in (0..18)] # Peter Luschny, May 19 2014

Formula

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.

a(n) ~ 5^(5*n)/(2^(7*n+4)*sqrt(Pi/10)*n^(3/2)). - Vaclav Kotesovec, Aug 20 2013

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

a(n) = 5*2^n*Gamma(5*n)/(Gamma(n)*Gamma(4*n+2)) for n > 0. - Peter Luschny, May 19 2014

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

G.f. A(x) satisfies: A(x) = 1 / (1 - 2 * x * A(x)^4). - Ilya Gutkovskiy, Nov 12 2021

Extensions

New definition and offset from Michael D. Weiner, Sep 03 2019