cp's OEIS Frontend

A053175 Catalan-Larcombe-French sequence.

%I A053175 #93 Aug 13 2024 04:06:53
%S A053175 1,8,80,896,10816,137728,1823744,24862720,346498048,4911669248,
%T A053175 70560071680,1024576061440,15008466534400,221460239482880,
%U A053175 3287994183188480,49074667327062016,735814252604162048
%N A053175 Catalan-Larcombe-French sequence.
%C A053175 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 A053175 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 A053175 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 A053175 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 A053175 T. D. Noe, <a href="/A053175/b053175.txt">Table of n, a(n) for n=0..200</a>
%H A053175 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 A053175 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 A053175 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 A053175 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 A053175 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 A053175 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 A053175 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 A053175 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 A053175 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 A053175 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 A053175 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 A053175 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 A053175 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 A053175 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 A053175 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 A053175 Zhi-Hong Sun, <a href="https://arxiv.org/abs/1803.10051">Congruences for Apéry-like numbers</a>, arXiv:1803.10051 [math.NT], 2018.
%H A053175 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 A053175 Yang Wen, <a href="http://sciencepublishinggroup.com/journal/paperinfo?journalid=616&amp;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 A053175 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 A053175 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 A053175 E.g.f.: exp(8*x)*BesselI(0, 4*x)^2. - _Vladeta Jovovic_, Aug 20 2003
%F A053175 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 A053175 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 A053175 a(n) ~ 2^(4n+1)/(Pi*n). - _Vaclav Kotesovec_, Oct 09 2012
%F A053175 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
%F A053175 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
%e A053175 G.f. = 1 + 8*x + 80*x^2 + 896*x^3 + 10816*x^4 + 137728*x^5 + 1823774*x^6 + ...
%p A053175 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 A053175 a[ n_] := SeriesCoefficient[ EllipticK[ (8 x /(1 - 8 x))^2] / ((1 - 8 x) Pi/2), {x, 0, n}]; (* _Michael Somos_, Aug 01 2011 *)
%t A053175 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 A053175 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 A053175 (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 A053175 (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 A053175 Cf. A065409, A002894, A081085.
%K A053175 nonn,nice
%O A053175 0,2
%A A053175 _Peter J Larcombe_, Nov 12 2001