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%I #85 Feb 27 2023 07:12:59
%S 1,8,80,896,10816,137728,1823744,24862720,346498048,4911669248,
%T 70560071680,1024576061440,15008466534400,221460239482880,
%U 3287994183188480,49074667327062016,735814252604162048
%N Catalan-Larcombe-French sequence.
%C 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.
%C 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
%D 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.
%D 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.
%H T. D. Noe, <a href="/A053175/b053175.txt">Table of n, a(n) for n=0..200</a>
%H E. Catalan, <a href="https://gdz.sub.uni-goettingen.de/id/PPN599472057_0001?tify={%22pages%22:[200]}">Sur les Nombres de Segner</a>, Rend. Circ. Mat. Pal., 1 (1887), 190-201. [From _Peter Luschny_, Jun 26 2009]
%H Lane Clark, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Clark/clark57.html">An asymptotic expansion for the Catalan-Larcombe-French sequence</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.1.
%H A. F. Jarvis, P. J. Larcombe and D. R. French, <a href="https://www.researchgate.net/publication/266565650_Linear_recurrences_between_two_recent_integer_sequences">Linear recurrences between two recent integer sequences</a>, Congressus Numerantium, 169 (2004), 79-99.
%H A. F. Jarvis, P. J. Larcombe and D. R. French, <a href="https://www.researchgate.net/publication/266172315_Applications_of_the_A_G_M_of_Gauss_some_new_properties_of_the_Catalan-Larcombe-French_sequence">Applications of the a.g.m. of Gauss: some new properties of the Catalan-Larcombe-French sequence</a>, Congressus Numerantium, 161 (2003), 151-162.
%H A. F. Jarvis, P. J. Larcombe and D. R. French, <a href="https://www.researchgate.net/publication/268889431_Power_series_identities_generated_by_two_recent_integer_sequences">Power series identities generated by two recent integer sequences</a>, Bulletin ICA, 43 (2005), 85-95.
%H A. F. Jarvis, P. J. Larcombe and D. R. French, <a href="https://www.researchgate.net/publication/265322144_On_small_prime_divisibility_of_the_Catalan-Larcombe-French_sequence">On Small Prime Divisibility of the Catalan-Larcombe-French sequence</a>, Indian Journal of Mathematics, 47 (2005), 159-181.
%H A. F. Jarvis, P. J. Larcombe and D. R. French, <a href="https://www.researchgate.net/publication/266565741_A_short_proof_of_the_2-adic_valuation_of_the_Catalan-Larcombe-French_number">A short proof of the 2-adic valuation of the Catalan-Larcombe-French number</a>, Indian Journal of Mathematics, 48 (2006), 135-138.
%H F. Jarvis, H. A. Verrill, <a href="https://doi.org/10.1007/s11139-009-9218-5">Supercongruences for the Catalan-Larcombe-French numbers</a>, Ramanujan J (22) (2010) 171.
%H Xiao-Juan Ji, Zhi-Hong Sun, <a href="http://arxiv.org/abs/1505.00668">Congruences for Catalan-Larcombe-French numbers</a>, arXiv:1505.00668 [math.NT], 2015 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Ji/ji6.html">JIS</a> vol 19 (2016) # 16.3.4
%H P. J. Larcombe, <a href="https://www.researchgate.net/publication/266573699_A_new_asymptotic_relation_between_two_recent_integer_sequences">A new asymptotic relation between two recent integer sequences</a>, Congressus Numerantium, 175 (2005), 111-116.
%H Peter J. Larcombe, Daniel R. French, <a href="https://www.researchgate.net/publication/268646122_On_the_other_Catalan_numbers_A_historical_formulation_re-examined">On the “Other” Catalan Numbers: A Historical Formulation Re-Examined</a>, Congressus Numerantium, 143 (2000), 33-64.
%H P. J. Larcombe and D. R. French, <a href="https://www.researchgate.net/publication/265702578_On_the_integrality_of_the_Catalan-Larcombe-French_sequence_188089610816">On the integrality of the Catalan-Larcombe-French sequence {1, 8, 80, 896, 10816, ...}</a>, Congressus Numerantium, 148 (2001), 65-91.
%H P. J. Larcombe and D. R. French, <a href="https://www.researchgate.net/publication/268890743_A_new_generating_function_for_the_Catalan-Larcombe-French_sequence_proof_of_a_result_by_Jovovic">A new generating function for the Catalan-Larcombe-French sequence: proof of a result by Jovovic</a>, Congressus Numerantium, 166 (2004), 161-172.
%H Guo-Shuai Mao, <a href="http://arxiv.org/abs/1511.06222">Proof of two supercongruences conjectured by Z.-W.Sun involving Catalan-Larcombe-French numbers</a>, arXiv:1511.06222 [math.NT], 2015.
%H Brian Yi Sun, Baoyindureng Wu, <a href="http://arxiv.org/abs/1602.04909">Two-log-convexity of the Catalan-Larcombe-French sequence</a>, arXiv:1602.04909 [math.CO], 2016. Also Journal of Inequalities and Applications, 2015, 2015:404; DOI: 10.1186/s13660-015-0920-0.
%H Zhi-Hong Sun, <a href="https://arxiv.org/abs/1803.10051">Congruences for Apéry-like numbers</a>, arXiv:1803.10051 [math.NT], 2018.
%H N. M. Temme, <a href="https://doi.org/10.1142/9789814612166_0013">Examples of 3_F_2-polynomials</a>, Asymptotic Methods for Integrals, Chapter 13, pp. 167-179 (2014).
%H Yang Wen, <a href="http://sciencepublishinggroup.com/journal/paperinfo?journalid=616&paperId=10018853">On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers</a>, American Journal of Mathematical and Computer Modelling, 2017; 2(4): 95-98.
%H E. X. W. Xia and O. X. M. Yao, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p3">A Criterion for the Log-Convexity of Combinatorial Sequences</a>, The Electronic Journal of Combinatorics, 20 (2013), #P3.
%F 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
%F E.g.f.: exp(8*x)*BesselI(0, 4*x)^2. - _Vladeta Jovovic_, Aug 20 2003
%F 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
%F 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
%F a(n) ~ 2^(4n+1)/(Pi*n). - _Vaclav Kotesovec_, Oct 09 2012
%F a(n) = 2^n*sum_{k=0}^n C(n,k)*C(2k,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,2k)*C(2k,k)^2*4^(n-2k). - _Zhi-Wei Sun_, Mar 21 2013
%F a(n) = (-1)^n*Sum_{k=0..n}C(2k,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
%e G.f. = 1 + 8*x + 80*x^2 + 896*x^3 + 10816*x^4 + 137728*x^5 + 1823774*x^6 + ...
%p 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]
%t a[ n_] := SeriesCoefficient[ EllipticK[ (8 x /(1 - 8 x))^2] / ((1 - 8 x) Pi/2), {x, 0, n}]; (* _Michael Somos_, Aug 01 2011 *)
%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ 8 x] BesselI[ 0, 4 x]^2, {x, 0, n}]]; (* _Michael Somos_, Aug 01 2011 *)
%t Table[(-8)^n Sqrt[Pi] HypergeometricPFQRegularized[{1/2, -n, -n}, {1, 1/2 - n}, -1]/n!, {n, 0, 20}] (* _Vladimir Reshetnikov_, May 21 2016 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / agm( 1, 1 - 16*x + x * O(x^n)), n))}; /* _Michael Somos_, Feb 12 2003 */
%o (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 */
%Y Cf. A065409, A002894, A081085.
%K nonn,nice
%O 0,2
%A _Peter J Larcombe_, Nov 12 2001
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