A038112 a(n) = T(2n,n), where T(n,k) is in A037027.

1, 2, 9, 40, 190, 924, 4578, 22968, 116325, 593450, 3045185, 15699840, 81260816, 421993040, 2197653240, 11472991008, 60023749566, 314621200260, 1651883008050, 8685998428800, 45734484854520, 241098942106440, 1272406536645660

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

Sequence in context: A021001 A231134 A370479 * A268039 A367044 A235596

Adjacent sequences: A038109 A038110 A038111 * A038113 A038114 A038115

Keyword

nonn,easy

Author

Floor van Lamoen

Status

approved