A026302 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 0, s(2n) = n. Also a(n) = T(2n,n), where T is the array in A026300.

1, 2, 9, 44, 230, 1242, 6853, 38376, 217242, 1239980, 7123765, 41141916, 238637282, 1389206210, 8112107475, 47495492400, 278722764954, 1638970147188, 9654874654438, 56965811111240, 336590781348276, 1991357644501170

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1280
• D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.

Crossrefs

Bisection of A026307.

Programs

Maple

b:= proc(x, y) option remember; `if`(min(x, y)<0, 0,
     `if`(max(x, y)=0, 1, b(x-1, y)+b(x, y-1)+b(x-2, y+1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 28 2019

Mathematica

Table[Binomial[2*n, n]*Hypergeometric2F1[1/2 - n/2, -n/2, 2 + n, 4], {n, 0, 30}] (* Vaclav Kotesovec, Sep 17 2019 *)

PARI

A026300(n,k)={ if(n<0 || k < 0, return(0) ;) ; if(n<=1, 1, if(k==0, 1, sum(i=0,k/2, binomial(n,2*i+n-k)*(binomial(2*i+n-k,i)-binomial(2*i+n-k,i-1))) ;) ;) ; }
A026302(n)={ A026300(2*n,n) ; }
{ for(n=0,21, print(n," ",A026302(n))) ; } \\ R. J. Mathar, Oct 26 2006

Formula

a(n) = binomial(2*n,n)*hypergeom([ -n/2, 1/2 - n/2],[n+2],4). - Mark van Hoeij, Jun 02 2010

a(n) = (n + 1) * A006605(n). - Mark van Hoeij, Jul 02 2010

G.f. A(x)=(x*M(x))', where M(x)=1+x*M(x)^2+x^2*M(x)^4. - Vladimir Kruchinin, May 25 2012

From Ilya Gutkovskiy, Sep 21 2017: (Start)

a(n) = [x^n] ((1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2))^(n+1).

a(n) = [x^n] (1/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - ...)))))))^(n+1), a continued fraction. (End)

From Vaclav Kotesovec, Sep 17 2019: (Start)

Recurrence: 3*n^2*(3*n + 1)*(3*n + 2)*(13*n - 9)*a(n) = 2*(n+1)*(2*n - 1)*(455*n^3 - 315*n^2 - 44*n + 24)*a(n-1) + 36*n*(n+1)*(2*n - 3)*(2*n - 1)*(13*n + 4)*a(n-2).

a(n) ~ sqrt(277 + 89*sqrt(13)) * (70 + 26*sqrt(13))^n / (13^(1/4) * sqrt(2*Pi*n) * 3^(3*n + 5/2)). (End)

Extensions

Corrected by R. J. Mathar, Oct 26 2006