A181418 a(n) = A000984(n)*A000172(n), which is the term-wise product of the Central binomial coefficients and Franel numbers, respectively.

1, 4, 60, 1120, 24220, 567504, 14030016, 360222720, 9513014940, 256758913840, 7051260776560, 196403499277440, 5535202897806400, 157551884911456000, 4522682234563776000, 130783762623673221120, 3806221127760278029980

Offset

0,2

Comments

This sequence is s_6 in Cooper's paper. - Jason Kimberley, Nov 25 2012

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

Links

Jason Kimberley, Table of n, a(n) for n = 0..226

S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).

Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.

Formula

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

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

From Jason Kimberley, Nov 26 2012: (Start)

1/Pi

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

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

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

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

a(n) ~ 2^(5*n+1) / (sqrt(3) * (Pi*n)^(3/2)). - Vaclav Kotesovec, Mar 06 2014

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

Example

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

Mathematica

Table[Binomial[2n, n]*Sum[Binomial[n, k]^3, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 06 2014 *)

Prog

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

Crossrefs

Cf. A000984, A000172, A199813.

Related to diagonal of rational functions: A268545-A268555.

Sequence in context: A123480 A227528 A156090 * A208890 A370498 A325154

Adjacent sequences: A181415 A181416 A181417 * A181419 A181420 A181421

Keyword

nonn,easy

Author

Paul D. Hanna, Jan 28 2011

Status

approved