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

0,2

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.

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)

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 *)

Prog

(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

Crossrefs

Bisection of A026307.

Sequence in context: A162356 A364476 A339440 * A214460 A124889 A317134

Adjacent sequences: A026299 A026300 A026301 * A026303 A026304 A026305

Keyword

nonn

Author

Clark Kimberling

Extensions

Corrected by R. J. Mathar, Oct 26 2006

Status

approved