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A053175
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Catalan-Larcombe-French sequence.
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6
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1, 8, 80, 896, 10816, 137728, 1823744, 24862720, 346498048, 4911669248, 70560071680, 1024576061440, 15008466534400, 221460239482880, 3287994183188480, 49074667327062016, 735814252604162048
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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.
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REFERENCES
| E. Catalan, Sur les Nombres de Segner, Rend. Circ. Mat. Pal., 1 (1887), 190-201. [From Peter Luschny (peter(AT)luschny.de), Jun 26 2009]
A. F. Jarvis, P. J. Larcombe and D. R. French, Linear recurrences between two recent integer sequences, Congressus Numerantium, 169 (2004), 79-99.
A. F. Jarvis, P. J. Larcombe and D. R. French, Applications of the a.g.m. of Gauss: some new properties of the Catalan-Larcombe-French sequence, Congressus Numerantium, 161 (2003), 151-162.
A. F. Jarvis, P. J. Larcombe and D. R. French, Power series identities generated by two recent integer sequences, Bulletin ICA, 43 (2005), 85-95.
A. F. Jarvis, P. J. Larcombe and D. R. French, On Small Prime Divisibility of the Catalan-Larcombe-French sequence, Indian Journal of Mathematics, 47 (2005), 159-181.
A. F. Jarvis, P. J. Larcombe and D. R. French, A short proof of the 2-adic valuation of the Catalan-Larcombe-French number, Indian Journal of Mathematics, 48 (2006), 135-138.
P. J. Larcombe, A new asymptotic relation between two recent integer sequences, Congressus Numerantium, 175 (2005), 111-116.
P. J. Larcombe and D. R. French, On the "other" Catalan numbers: a historical formulation re-examined, Congressus Numerantium, 143 (2000), 33-64.
P. J. Larcombe and D. R. French, On the integrality of the Catalan-Larcombe-French sequence {1, 8, 80, 896, 10816, ...}, Congressus Numerantium, 148 (2001), 65-91.
P. J. Larcombe and D. R. French, A new generating function for the Catalan-Larcombe-French sequence: proof of a result by Jovovic, Congressus Numerantium, 166 (2004), 161-172.
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.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
Lane Clark, An asymptotic expansion for the Catalan-Larcombe-French sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.1.
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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 (vladeta(AT)eunet.rs), Dec 30, 2003
E.g.f.: exp(8*x)*BesselI(0, 4*x)^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), 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 (vladeta(AT)eunet.rs), Sep 09 2003
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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; [From Peter Luschny, Jun 26 2009]
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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 *)
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PROG
| (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 */
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CROSSREFS
| Cf. A065409, A002894, A081085.
Sequence in context: A155144 A136949 A102592 * A051580 A060375 A097815
Adjacent sequences: A053172 A053173 A053174 * A053176 A053177 A053178
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KEYWORD
| nonn,nice
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AUTHOR
| P.J.Larcombe(AT)derby.ac.uk, Nov 12 2001
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