A174123 - OEIS

%I #28 Jan 12 2025 15:56:57

%S 1,4,19,112,751,5404,40573,313408,2471167,19791004,160459069,

%T 1313922064,10847561089,90174127684,754009158019,6336733626112,

%U 53489159252671,453258909448636,3854034482891725,32871004555812112,281127047928811201,2410285909684370788

%N Partial sums of A002893.

%H Tewodros Amdeberhan and Roberto Tauraso, <a href="http://arxiv.org/abs/1607.02483">Two triple binomial sum supercongruences</a>, arXiv:1607.02483 [math.NT], Jul 08 2016.

%F a(n) = Sum_{i=0..n} A002893(i).

%F From _Sergey Perepechko_, Feb 16 2011: (Start)

%F O.g.f.: 2*sqrt(2)/Pi/(1-z)/sqrt(1-6*z-3*z^2+sqrt((1-z)^3*(1-9*z)))* EllipticK(8*z^(3/2)/(1-6*z-3*z^2+sqrt((1-z)^3*(1-9*z)))).

%F 9*(n+2)^2*a(n) - (99+86*n+19*n^2)*a(n+1) + (72+56*n+11*n^2)*a(n+2) - (n+3)^2*a(n+3)=0. (End)

%F a(n) ~ 3^(2*n + 7/2) / (32*Pi*n). - _Vaclav Kotesovec_, Jul 11 2016

%t Accumulate[Table[Sum[Binomial[n,k]^2 Binomial[2k,k],{k,0,n}],{n,0,20}]] (* _Harvey P. Dale_, May 05 2013 *)

%o (PARI) a(n)=sum(m=0,n,sum(k=0,m,binomial(m,k)^2*binomial(2*k,k)))

%Y Cf. A002893, A000172, A002895, A000984, A207321.

%K easy,nonn

%O 0,2

%A _Jonathan Vos Post_, Mar 08 2010