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A181418 a(n) = A000984(n)*A000172(n), which is the term-wise product of the Central binomial coefficients and Franel numbers, respectively. 4
1, 4, 60, 1120, 24220, 567504, 14030016, 360222720, 9513014940, 256758913840, 7051260776560, 196403499277440, 5535202897806400, 157551884911456000, 4522682234563776000, 130783762623673221120, 3806221127760278029980 (list; graph; refs; listen; history; text; internal format)
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).

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.

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) - Jason Kimberley, Nov 26 2012

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 A013486 A013483

Adjacent sequences:  A181415 A181416 A181417 * A181419 A181420 A181421

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Jan 28 2011

STATUS

approved

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Last modified December 3 14:45 EST 2016. Contains 278745 sequences.