%I #38 Apr 04 2023 11:44:58

%S 1,4,60,1120,24220,567504,14030016,360222720,9513014940,256758913840,

%T 7051260776560,196403499277440,5535202897806400,157551884911456000,

%U 4522682234563776000,130783762623673221120,3806221127760278029980

%N a(n) = A000984(n)*A000172(n), which is the term-wise product of the Central binomial coefficients and Franel numbers, respectively.

%C This sequence is s_6 in Cooper's paper. - _Jason Kimberley_, Nov 25 2012

%C Diagonal of the rational function R(x,y,z,w)=1/(1-(w*x*y+w*z+x*y+x*z+y+z)). - _Gheorghe Coserea_, Jul 13 2016

%H Jason Kimberley, <a href="/A181418/b181418.txt">Table of n, a(n) for n = 0..226</a>

%H S. Cooper, <a href="http://dx.doi.org/10.1007/s11139-011-9357-3">Sporadic sequences, modular forms and new series for 1/pi</a>, Ramanujan J. (2012).

%H Timothy Huber, Daniel Schultz, and Dongxi Ye, <a href="https://doi.org/10.4064/aa220621-19-12">Ramanujan-Sato series for 1/pi</a>, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.

%F a(n) = C(2n,n) * Sum_{k=0..n} C(n,k)^3.

%F E.g.f.: Sum_{n>=0} a(n)*x^n/(n!*(2*n)!) = ( Sum_{n>=0} x^n/n!^3 )^2.

%F From _Jason Kimberley_, Nov 26 2012: (Start)

%F 1/Pi

%F = (2/25)*Sum_{n>=0} a(n)*(9n+2)/50^n. [Cooper, equation (5)]

%F = (2/25)*Sum_{n>=0} a(n)*A017185(n)/A165800(n). (End)

%F G.f.: 4*hypergeom([1/6, 1/3],[1],(27/2)*(1+(1-32*x)^(1/2))*(1-(1-32*x)^(1/2))^2/(3+(1-32*x)^(1/2))^3)^2/(3+(1-32*x)^(1/2)). - _Mark van Hoeij_, May 07 2013

%F Recurrence: n^3*a(n) = 2*(2*n-1)*(7*n^2 - 7*n + 2)*a(n-1) + 32*(n-1)*(2*n-3)*(2*n-1)*a(n-2). - _Vaclav Kotesovec_, Mar 06 2014

%F a(n) ~ 2^(5*n+1) / (sqrt(3) * (Pi*n)^(3/2)). - _Vaclav Kotesovec_, Mar 06 2014

%F 0 = (-x^2+28*x^3+128*x^4)*y''' + (-3*x+126*x^2+768*x^3)*y'' + (-1+92*x+864*x^2)*y' + (4+96*x)*y, where y is g.f. - _Gheorghe Coserea_, Jul 13 2016

%e E.g.f.: A(x) = 1 + 4*x/2! + 60*x^2/(2!*4!) + 1120*x^3/(3!*6!) + 24220*x^4/(4!*8!) + 567504*x^5/(5!*10!) +....

%e where A(x)^(1/2) = 1 + x + x^2/2!^3 + x^3/3!^3 + x^4/4!^3 +x^5/5!^3 +...

%t Table[Binomial[2n,n]*Sum[Binomial[n,k]^3,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Mar 06 2014 *)

%o (PARI) {a(n)=binomial(2*n,n)*sum(k=0,n,binomial(n,k)^3)}

%o (PARI) {a(n)=(2*n)!*n!*polcoeff(sum(m=0, n, x^m/m!^3+x*O(x^n))^2, n)}

%Y Cf. A000984, A000172, A199813.

%Y Related to diagonal of rational functions: A268545-A268555.

%K nonn,easy

%O 0,2

%A _Paul D. Hanna_, Jan 28 2011