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A038112 a(n) = T(2n,n), where T(n,k) is in A037027. 6
1, 2, 9, 40, 190, 924, 4578, 22968, 116325, 593450, 3045185, 15699840, 81260816, 421993040, 2197653240, 11472991008, 60023749566, 314621200260, 1651883008050, 8685998428800, 45734484854520, 241098942106440, 1272406536645660 (list; graph; refs; listen; history; text; internal format)
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: A107979 A021001 A231134 * A268039 A235596 A052512

Adjacent sequences:  A038109 A038110 A038111 * A038113 A038114 A038115

KEYWORD

nonn,easy

AUTHOR

Floor van Lamoen

STATUS

approved

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Last modified February 22 05:54 EST 2020. Contains 332116 sequences. (Running on oeis4.)