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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A053175 Catalan-Larcombe-French sequence.

Original entry on oeis.org

1, 8, 80, 896, 10816, 137728, 1823744, 24862720, 346498048, 4911669248, 70560071680, 1024576061440, 15008466534400, 221460239482880, 3287994183188480, 49074667327062016, 735814252604162048
Offset: 0

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Author

Peter J Larcombe, Nov 12 2001

Keywords

Comments

These numbers were proposed as 'Catalan' numbers by an associate of Catalan. They appear as coefficients in the series expansion of an elliptic integral of the first kind. Defining f(x; c) = 1 /(1 - c^2*sin^2(x))^(1/2), consider the function I(c) obtained by integrating f(x; c) with respect to x between 0 and Pi/2. I(c) is transformed and written as a power series in c (through an intermediate variable) which acts as a generating function for the sequence.
Conjecture: Let P(n) be the (n+1) X (n+1) Hankel-type determinant with (i,j)-entry equal to a(i+j) for all i,j = 0,...,n. Then P(n)/2^(n*(n+3)) is a positive odd integer. - Zhi-Wei Sun, Aug 14 2013

Examples

			G.f. = 1 + 8*x + 80*x^2 + 896*x^3 + 10816*x^4 + 137728*x^5 + 1823774*x^6 + ...

References

  • P. J. Larcombe, D. R. French and E. J. Fennessey, The asymptotic behavior of the Catalan-Larcombe-French sequence {1, 8, 80, 896, 10816, ...}, Utilitas Mathematica, 60 (2001), 67-77.
  • P. J. Larcombe, D. R. French and C. A. Woodham, A note on the asymptotic behavior of a prime factor decomposition of the general Catalan-Larcombe-French number, Congressus Numerantium, 156 (2002), 17-25.

Links

Crossrefs

Cf. A065409, A002894, A081085.

Programs

  • Maple
    a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 8 else (8*(3*n^2 -3*n+1)*a(n-1)-128*(n-1)^2*a(n-2))/n^2 fi end; # Peter Luschny, Jun 26 2009
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticK[ (8 x /(1 - 8 x))^2] / ((1 - 8 x) Pi/2), {x, 0, n}]; (* Michael Somos, Aug 01 2011 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ 8 x] BesselI[ 0, 4 x]^2, {x, 0, n}]]; (* Michael Somos, Aug 01 2011 *)
Table[(-8)^n Sqrt[Pi] HypergeometricPFQRegularized[{1/2, -n, -n}, {1, 1/2 - n}, -1]/n!, {n, 0, 20}] (* Vladimir Reshetnikov, May 21 2016 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( 1 / agm( 1, 1 - 16*x + x * O(x^n)), n))}; /* Michael Somos, Feb 12 2003 */
  • PARI
    {a(n) = if( n<0, 0, polcoeff( sum( k=0, n, binomial( 2*k ,k)^2 * (2*x - 16*x^2)^k, x * O(x^n)), n))}; /* Michael Somos, Mar 04 2003 */

Formula

G.f.: 1 / AGM(1, 1 - 16*x) = 2 * EllipticK(8*x / (1-8*x)) / ((1-8*x)*Pi), where AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre. Cf. A081085, A089602. - Michael Somos, Mar 04 2003 and Vladeta Jovovic, Dec 30 2003
E.g.f.: exp(8*x)*BesselI(0, 4*x)^2. - Vladeta Jovovic, Aug 20 2003
a(n)*n^2 = a(n-1)*8*(3*n^2 - 3*n + 1) - a(n-2)*128*(n-1)^2. - Michael Somos, Apr 01 2003
Exponential convolution of A059304 with itself: Sum(2^n*binomial(2*n, n)*x^n/n!, n=0..infinity)^2 = (BesselI(0, 4*x)*exp(4*x))^2 = hypergeom([1/2], [1], 8*x)^2. - Vladeta Jovovic, Sep 09 2003
a(n) ~ 2^(4n+1)/(Pi*n). - Vaclav Kotesovec, Oct 09 2012
a(n) = 2^n*Sum_{k=0..n} C(n,k)*C(2*k,k)*C(2(n-k),n-k), where C(n,k)=n!/(k!*(n-k)!). This formula has been proved via the Zeilberger algorithm (both sides of the equality satisfy the same recurrence relation). a(n)/2^n also has another expression: Sum_{k=0..floor(n/2)} C(n,2*k)*C(2*k,k)^2*4^(n-2*k). - Zhi-Wei Sun, Mar 21 2013
a(n) = (-1)^n*Sum_{k=0..n}C(2*k,k)*C(2(n-k),n-k)*C(k,n-k)*(-4)^k. I have proved this new formula via the Zeilberger algorithm. - Zhi-Wei Sun, Nov 19 2014

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