A120278 a(n) = Sum_{m=1..n} Sum_{k=1..m} C(2*k,k), where C(2*k,k) = (2*k)!/(k!)^2 = A000984(k).

2, 10, 38, 136, 486, 1760, 6466, 24042, 90238, 341190, 1297574, 4958114, 19019254, 73196994, 282492254, 1092867904, 4236849774, 16455966944, 64020347914, 249431257704, 973100041934, 3800867789884, 14862066265434, 58170868424084

Offset: 1

Author

Alexander Adamchuk, Jul 04 2006

Keywords

• nonn
• changed

Comments

a(2*(p-1)) is divisible by p^2 for p=7,13,19,31,37,43,61,67.. A002476 (Primes of the form 6m + 1).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A000984, A066796, A002476.

Cf. A066796, A079309.

Programs

Mathematica

Table[Sum[Sum[(2k)!/(k!)^2,{k,1,m}],{m,1,n}],{n,1,50}]
CoefficientList[Series[(1/Sqrt[1-4 x]-1)/((x-1)^2 x),{x,0,50}],x] (* Harvey P. Dale, May 24 2011 *)

Formula

a(n) = Sum_{m=1..n} Sum_{k=1..m} (2*k)!/(k!)^2.

a(n) = 2 * Sum_{k=1..n} A079309(k) = Sum_{k=1..n} A066796(k). - Alexander Adamchuk, Sep 01 2006

G.f.: x*(1/sqrt(1-4*x)-1)/(x*(x-1)^2). - Harvey P. Dale, May 24 2011

Recurrence: n*a(n) = 2*(3*n-1)*a(n-1) - (9*n-4)*a(n-2) + 2*(2*n-1)*a(n-3). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ 2^(2*n+4)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012