A137644 a(n) = Sum_{k=0..n} C(n+k,k)*C(n+k,n-k).

1, 3, 16, 95, 591, 3780, 24620, 162423, 1081780, 7258053, 48982176, 332140328, 2261099491, 15444137880, 105789736896, 726426836103, 4998885106599, 34464824536500, 238017084356680, 1646234203000485, 11401464090042224, 79060352485691272, 548829398923188036

Offset: 0

Author

Paul D. Hanna, Jan 31 2008

Keywords

• nonn

Comments

Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (0,1), (0,2). - Eric Werley, Jun 29 2011

Diagonal of rational function 1/(1 - (x + y + x*y + x^2)). - Gheorghe Coserea, Aug 31 2018

Examples

			The triangle of number of lattice paths from (0,0) to (n,k) using steps (1,0), (1,1), (0,1), (0,2) begins:
  1;
  1,  3;
  1,  5,  16;
  1,  7,  29,  95;
  1,  9,  46, 179,  591;
  1, 11,  67, 303, 1140,  3780;
  1, 13,  92, 475, 2010,  7405, 24620;
  1, 15, 121, 703, 3309, 13427, 48761, 162423;
  1, 17, 154, 995, 5161, 22892, 90241, 324317, 1081780;
This sequence is the diagonal. - _Joerg Arndt_, Jul 01 2011
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A063967.

Programs

Mathematica

Table[ HypergeometricPFQ[{-n, 1 + n, 1 + n}, {1/2, 1}, -(1/4)], {n,0,20}] (* Olivier Gérard, Apr 23 2009 *)
Table[Sum[Binomial[n+k,k]Binomial[n+k,n-k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Aug 03 2011 *)

PARI

a(n)=sum(k=0,n,binomial(n+k,k)*binomial(n+k,n-k))

PARI

/* same as in A092566 but use */
steps=[[1,0], [1,1], [0,1], [0,2]];
/* Joerg Arndt, Jun 30 2011 */

Formula

a(n) = 3F2( {-n, n+1, n+1}; {1/2, 1})( -(1/4) ). - Olivier Gérard, Apr 23 2009

G.f.: F'(x)/(1+F(x)), where F(x)=x*(1+F(x))/(1-F(x)-F(x)^2). - Vladimir Kruchinin, Mar 24 2012

a(n) = A063967(n,n). - Alois P. Heinz, Oct 11 2017

a(n) ~ sqrt(56 + (7*(15953 - 267*sqrt(105)))^(1/3) + (7*(15953 + 267*sqrt(105)))^(1/3)) * (((36 + (44766 - 1050*sqrt(105))^(1/3) + (6*(7461 + 175*sqrt(105)))^(1/3))/15)^n / sqrt(210*Pi*n)). - Vaclav Kotesovec, Feb 17 2024