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%I #59 Aug 10 2018 02:49:31
%S 1,2,9,40,190,924,4578,22968,116325,593450,3045185,15699840,81260816,
%T 421993040,2197653240,11472991008,60023749566,314621200260,
%U 1651883008050,8685998428800,45734484854520,241098942106440,1272406536645660
%N a(n) = T(2n,n), where T(n,k) is in A037027.
%C Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (2,0). - _Joerg Arndt_, Jun 30 2011
%C Diagonal of rational function 1/(1 - (x + y + y^2)). - _Gheorghe Coserea_, Aug 06 2018
%H Vincenzo Librandi, <a href="/A038112/b038112.txt">Table of n, a(n) for n = 0..1000</a>
%H P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry3/barry132.html">On the Central Coefficients of Bell Matrices</a>, J. Int. Seq. 14 (2011) # 11.4.3 example 7.
%F a(n) = Sum_{k=0..n} C(n+k,k)*C(k,n-k). - _Paul Barry_, May 13 2006
%F a(n) = Sum_{j=0..n} binomial(2*j, n)*binomial(n+j, 2*j). - _Zerinvary Lajos_, Aug 22 2006
%F a(n) = [x^n] (1/(1-x-x^2))^(n+1). - _Paul Barry_, Mar 23 2011
%F a(n) = (n+1)*A001002(n+1).
%F G.f.: Sum_{n>=0} d^n/dx^n x^(2*n)*(1+x)^n/n!. - _Paul D. Hanna_, Aug 04 2012
%F Recurrence: 5*(n-1)*n*a(n) = 11*(n-1)*(2*n-1)*a(n-1) + 3*(3*n-4)*(3*n-2)*a(n-2). - _Vaclav Kotesovec_, Oct 08 2012
%F a(n) ~ 3^(3*n+3/2)/(2^(3/2)*5^(n+1/2)*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 08 2012
%F G.f.: A(x) where (x+1)*(27*x-5)*A(x)^3 + 4*A(x) + 1 = 0. - _Mark van Hoeij_, May 01 2013
%e G.f.: A(x) = 1 + 2*x + 9*x^2 + 40*x^3 + 190*x^4 + 924*x^5 + 4578*x^6 + ...
%p a:=n->sum(binomial(2*j,n)*(binomial(n+j,2*j)),j=0..n): seq(a(n), n=0..21); # _Zerinvary Lajos_, Aug 22 2006
%p series(RootOf((x+1)*(27*x-5)*A^3+4*A+1,A),x=0,30); # _Mark van Hoeij_, May 01 2013
%t Table[Sum[Binomial[n+k,k]Binomial[k,n-k],{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Sep 30 2012 *)
%t Table[Binomial[2 n, n] Hypergeometric2F1[1/2 - n/2, -n/2, -2 n, -4], {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 19 2016 *)
%o (PARI) {a(n) = if( n<0, 0, sum(k=0, n\2, (2*n-k)!/ (k! * (n-2*k)!)) / n!)}; /* _Michael Somos_, Sep 29 2003 */
%o (PARI) {a(n) = if( n<0, 0, n++; n * polcoeff(serreverse( x - x^2 - x^3 + x * O(x^n)), n))}; /* _Michael Somos_, Sep 29 2003 */
%o (PARI) /* same as in A092566 but use */
%o steps=[[0,1], [1,0], [2,0]]; /* _Joerg Arndt_, Jun 30 2011 */
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} \\ = d^n/dx^n F
%o {a(n)=local(A=x); A=1+sum(m=1, n, Dx(m, x^(2*m)*(1+x+x*O(x^n))^m/m!)); polcoeff(A, n)} \\ _Paul D. Hanna_, Aug 04 2012
%o (GAP) List([0..25],n->Sum([0..n],k->Binomial(n+k,k)*Binomial(k,n-k))); # _Muniru A Asiru_, Aug 06 2018
%K nonn,easy
%O 0,2
%A _Floor van Lamoen_
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