A203611 a(n) = Sum_{k=0..n} binomial(k-1,2*k-1-n)*binomial(k,2*k-n), with a(0) = 1.

1, 1, 1, 3, 7, 16, 39, 95, 233, 577, 1436, 3590, 9011, 22691, 57299, 145043, 367931, 935078, 2380405, 6068745, 15492702, 39598631, 101323446, 259522398, 665332007, 1707137941, 4383662419, 11264675925, 28966161253, 74530441162, 191879611399, 494265165151

Offset: 0

Author

Peter Luschny, Jan 14 2012

Keywords

• nonn

Comments

For the connection with Fibonacci meanders classified by maximal run length of 1s (see the link).

Apparently the number of grand Motzkin paths of length n+1 that avoid UU. - David Scambler, Jul 04 2013

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2397
• Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, Grand Dyck paths with air pockets, arXiv:2211.04914 [math.CO], 2022.
• Jean-Luc Baril and José L. Ramírez, Fibonacci and Catalan paths in a wall, 2023.
• Peter Luschny, Fibonacci meanders.

Crossrefs

Cf. A110236, bisection of A202411.

Programs

Magma

A203611:= func< n | n eq 0 select 1 else (&+[Binomial(k-1,2*k-n-1)*Binomial(k,2*k-n): k in [0..n]]) >;
[A203611(n): n in [0..40]]; // G. C. Greubel, Mar 12 2025

Maple

a := n -> hypergeom([-n/2, 1 - n/2, (1-n)/2, (1-n)/2], [1, -n, 1 - n], 16):
seq(simplify(a(n)), n = 0..31); # Peter Luschny, Mar 24 2023

Mathematica

a[n_] := Module[{a, r, b, c, d, z}, If[n == 0, Return[1]]; a = Quotient[n, 2]; r = n-1; b = a-r/2+1; c = a+1; d = a-r; z = If[Mod[n, 2] == 1, (n+1)/2, n^2*(n+2)/16]; z*HypergeometricPFQ[{1, c, c+1, d, d}, {b, b, b-1/2, b+1/2}, 1/16] ]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jun 27 2013, translated from Maple *)
Table[Sum[Binomial[k-1,2k-1-n]Binomial[k,2k-n],{k,0,n}],{n,0,40}] (* Harvey P. Dale, May 25 2014 *)

PARI

my(x='x+O('x^66)); Vec( 2*x/((1+x-x^2) * sqrt((x^2+x+1) * (x^2-3*x+1)) -x^4 +2*x^3 +x^2 +2*x -1) ) \\ Joerg Arndt, May 06 2013

SageMath

def A203611(n): return 1 if n==0 else sum(binomial(k-1,2*k-n-1)*binomial(k,2*k-n) for k in range(n+1))
print([A203611(n) for n in range(41)]) # G. C. Greubel, Mar 12 2025

Formula

For n>0 let A=floor(n/2), R = n-1, B = A - R/2 + 1, C = A+1, D = A-R and Z = (n+1)/2 if n mod 2 = 1, otherwise Z = n^2*(n+2)/16. Then a(n) = Z*Hypergeometric5F4([1,C,C+1,D,D],[B,B,B-1/2,B+1/2],1/16).

G.f.: 2*x/((1+x-x^2)*sqrt((1+x+x^2)*(1-3*x+x^2)) - (1-2*x-x^2-2*x^3+x^4)). - Mark van Hoeij, May 06 2013

a(n) ~ phi^(2*n + 1) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 08 2019

a(n) = hypergeom([-n/2, 1 - n/2, (1-n)/2, (1-n)/2], [1, -n, 1 - n], 16). - Peter Luschny, Mar 24 2023

D-finite with recurrence n*a(n) -(n+1)*a(n-1) -2*(2*n-5)*a(n-2) -(n+3)*a(n-3) +3*(n-5)*a(n-5) -(n-6)*a(n-6) = 0. - R. J. Mathar, Nov 22 2024