OFFSET
0,2
COMMENTS
Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (2,0). - Joerg Arndt, Jun 30 2011
Diagonal of rational function 1/(1 - (x + y + y^2)). - Gheorghe Coserea, Aug 06 2018
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
P. Barry, On the Central Coefficients of Bell Matrices, J. Int. Seq. 14 (2011) # 11.4.3 example 7.
FORMULA
a(n) = Sum_{k=0..n} C(n+k,k)*C(k,n-k). - Paul Barry, May 13 2006
a(n) = Sum_{j=0..n} binomial(2*j, n)*binomial(n+j, 2*j). - Zerinvary Lajos, Aug 22 2006
a(n) = [x^n] (1/(1-x-x^2))^(n+1). - Paul Barry, Mar 23 2011
a(n) = (n+1)*A001002(n+1).
G.f.: Sum_{n>=0} d^n/dx^n x^(2*n)*(1+x)^n/n!. - Paul D. Hanna, Aug 04 2012
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
a(n) ~ 3^(3*n+3/2)/(2^(3/2)*5^(n+1/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012
G.f.: A(x) where (x+1)*(27*x-5)*A(x)^3 + 4*A(x) + 1 = 0. - Mark van Hoeij, May 01 2013
EXAMPLE
G.f.: A(x) = 1 + 2*x + 9*x^2 + 40*x^3 + 190*x^4 + 924*x^5 + 4578*x^6 + ...
MAPLE
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
series(RootOf((x+1)*(27*x-5)*A^3+4*A+1, A), x=0, 30); # Mark van Hoeij, May 01 2013
MATHEMATICA
Table[Sum[Binomial[n+k, k]Binomial[k, n-k], {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Sep 30 2012 *)
Table[Binomial[2 n, n] Hypergeometric2F1[1/2 - n/2, -n/2, -2 n, -4], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 19 2016 *)
PROG
(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 */
(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 */
(PARI) /* same as in A092566 but use */
steps=[[0, 1], [1, 0], [2, 0]]; /* Joerg Arndt, Jun 30 2011 */
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} \\ = d^n/dx^n F
{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
(GAP) List([0..25], n->Sum([0..n], k->Binomial(n+k, k)*Binomial(k, n-k))); # Muniru A Asiru, Aug 06 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved