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%I #46 Dec 01 2023 03:24:55
%S 1,2,9,44,230,1242,6853,38376,217242,1239980,7123765,41141916,
%T 238637282,1389206210,8112107475,47495492400,278722764954,
%U 1638970147188,9654874654438,56965811111240,336590781348276,1991357644501170
%N 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.
%H Alois P. Heinz, <a href="/A026302/b026302.txt">Table of n, a(n) for n = 0..1280</a>
%H D. Kruchinin and V. Kruchinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kruchinin/kruchinin5.html">A Method for Obtaining Generating Function for Central Coefficients of Triangles</a>, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.
%F a(n) = binomial(2*n,n)*hypergeom([ -n/2, 1/2 - n/2],[n+2],4). - _Mark van Hoeij_, Jun 02 2010
%F a(n) = (n + 1) * A006605(n). - _Mark van Hoeij_, Jul 02 2010
%F 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
%F From _Ilya Gutkovskiy_, Sep 21 2017: (Start)
%F a(n) = [x^n] ((1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2))^(n+1).
%F 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)
%F From _Vaclav Kotesovec_, Sep 17 2019: (Start)
%F 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).
%F 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)
%p b:= proc(x, y) option remember; `if`(min(x, y)<0, 0,
%p `if`(max(x, y)=0, 1, b(x-1, y)+b(x, y-1)+b(x-2, y+1)))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..23); # _Alois P. Heinz_, Sep 28 2019
%t Table[Binomial[2*n, n]*Hypergeometric2F1[1/2 - n/2, -n/2, 2 + n, 4], {n, 0, 30}] (* _Vaclav Kotesovec_, Sep 17 2019 *)
%o (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))) ;) ;) ; }
%o A026302(n)={ A026300(2*n,n) ; }
%o { for(n=0,21, print(n," ",A026302(n))) ; } \\ _R. J. Mathar_, Oct 26 2006
%Y Bisection of A026307.
%K nonn
%O 0,2
%A _Clark Kimberling_
%E Corrected by _R. J. Mathar_, Oct 26 2006
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