A089602 -id:A089602 - cp's OEIS frontend

A053175 Catalan-Larcombe-French sequence.

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

Offset: 0

Peter J Larcombe, Nov 12 2001

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

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

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.

T. D. Noe, Table of n, a(n) for n=0..200
E. Catalan, Sur les Nombres de Segner, Rend. Circ. Mat. Pal., 1 (1887), 190-201. [From _Peter Luschny_, Jun 26 2009]
Lane Clark, An asymptotic expansion for the Catalan-Larcombe-French sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.1.
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.
F. Jarvis, H. A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J (22) (2010) 171.
Xiao-Juan Ji, Zhi-Hong Sun, Congruences for Catalan-Larcombe-French numbers, arXiv:1505.00668 [math.NT], 2015 and JIS vol 19 (2016) # 16.3.4
P. J. Larcombe, A new asymptotic relation between two recent integer sequences, Congressus Numerantium, 175 (2005), 111-116.
Peter J. Larcombe, Daniel 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.
Guo-Shuai Mao, Proof of two supercongruences conjectured by Z.-W.Sun involving Catalan-Larcombe-French numbers, arXiv:1511.06222 [math.NT], 2015.
Brian Yi Sun, Baoyindureng Wu, Two-log-convexity of the Catalan-Larcombe-French sequence, arXiv:1602.04909 [math.CO], 2016. Also Journal of Inequalities and Applications, 2015, 2015:404; DOI: 10.1186/s13660-015-0920-0.
Zhi-Hong Sun, Congruences for Apéry-like numbers, arXiv:1803.10051 [math.NT], 2018.
N. M. Temme, Examples of 3_F_2-polynomials, Asymptotic Methods for Integrals, Chapter 13, pp. 167-179 (2014).
Yang Wen, On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers, American Journal of Mathematical and Computer Modelling, 2017; 2(4): 95-98.
E. X. W. Xia and O. X. M. Yao, A Criterion for the Log-Convexity of Combinatorial Sequences, The Electronic Journal of Combinatorics, 20 (2013), #P3.

Cf. A065409, A002894, A081085.

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

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 *)

{a(n) = if( n<0, 0, polcoeff( 1 / agm( 1, 1 - 16*x + x * O(x^n)), n))}; /* Michael Somos, Feb 12 2003 */

{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 */

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

A089603 G.f.: sqrt(1/agm(1, 1-8*x)) = sqrt(o.g.f. for A081085).

Original entry on oeis.org

1, 2, 8, 40, 226, 1380, 8880, 59280, 406416, 2842400, 20186752, 145119616, 1053575336, 7711639760, 56834201280, 421327859520, 3139306406850, 23494847031300, 176526280319120, 1330929290036560, 10065855468854980, 76341682531733960, 580460500453098080

Offset: 0

N. J. A. Sloane, Dec 31 2003

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Arithmetic-geometric mean
Wikipedia, Arithmetic-geometric mean

Cf. A081085, A089602.

CoefficientList[Series[Sqrt[2*EllipticK[1/(1 - 1/(4*x))^2]/(Pi*(1 - 4*x))], {x, 0, 25}], x] (* Vaclav Kotesovec, Sep 26 2019 *)
nmax = 25; CoefficientList[Series[Sqrt[Hypergeometric2F1[1/2, 1/2, 1, 16*x*(1 - 4*x)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 26 2019 *)

Vec( 1/agm(1,1-8*x+O(x^66))^(1/2) ) \\ Joerg Arndt, Aug 14 2013

a(n) ~ 2^(3*n - 1/2) / (n * sqrt(Pi*log(n))) * (1 - (gamma/2 + log(2))/log(n) + (3*gamma^2/8 + 3*log(2)*gamma/2 + 3*log(2)^2/2 - Pi^2/16) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

A090004 Expansion of L(x)^(1/2), where L(x) is the g.f. for the Catalan Larcombe-French sequence A053175.

Original entry on oeis.org

1, 4, 32, 320, 3616, 44160, 568320, 7587840, 104042496, 1455308800, 20671234048, 297204973568, 4315444576256, 63173752913920, 931171553771520, 13806071300751360, 205737584679321600, 3079516590086553600, 46275305227975393280, 697790255614687969280, 10554814464110079508480

Offset: 0

Peter J Larcombe, Jan 19 2004

Seiichi Manyama, Table of n, a(n) for n = 0..833

Cf. A053175, A089602, A089603.

nmax = 25; CoefficientList[Series[(EllipticK[(8*x/(1 - 8*x))^2]/((1 - 8*x)*Pi/2))^(1/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 26 2019 *)

Vec( 1/agm(1,1-16*x+O(x^66))^(1/2) ) \\ Joerg Arndt, Aug 14 2013

a(n) = 2^n * A089603(n). - Seiichi Manyama, Jan 13 2019

a(n) ~ 2^(4*n - 1/2) / (n * sqrt(Pi*log(n))) * (1 - (gamma/2 + log(2))/log(n) + (3*gamma^2/8 + 3*log(2)*gamma/2 + 3*log(2)^2/2 - Pi^2/16) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

More terms from Christian G. Bower, Jan 19 2004

A323385 Expansion of AGM(1,1-16*x) (where AGM denotes the arithmetic-geometric mean).

Original entry on oeis.org

1, -8, -16, -128, -1344, -15872, -199680, -2613248, -35148800, -482500608, -6730072064, -95094702080, -1358152794112, -19573573681152, -284284750397440, -4156674357067776, -61133523873169408, -903754859816157184, -13421680957337894912, -200140704802846801920

Offset: 0

Seiichi Manyama, Jan 13 2019

sign

Robert Israel, Table of n, a(n) for n = 0..833
Eric Weisstein's World of Mathematics, Arithmetic-geometric mean
Wikipedia, Arithmetic-geometric mean

Convolution inverse of A053175.

Cf. A060691, A089602, A090004.

R:= Pi*(1-8*x)/(2*EllipticK(-8*x/(1-8*x))):
S:= series(R,x,31):
seq(coeff(S,x,j),j=0..30); # Robert Israel, Jan 13 2019

CoefficientList[Series[Pi*(1 - 16*x) / (2*EllipticK[1 - 1/(1 - 16*x)^2]), {x, 0, 25}], x] (* Vaclav Kotesovec, Sep 28 2019 *)

PARI
```
N=66; x='x+O('x^N); Vec(agm(1, 1-16*x))
```

a(n) = 2^n * A060691(n).

a(n) ~ -Pi * 2^(4*n-1) / (n * log(n)^2) * (1 - (2*gamma + 4*log(2))/log(n) + (3*gamma^2 + 12*log(2)*gamma + 12*log(2)^2 - Pi^2/2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

A228156 Expansion of sqrt((1+4x)/AGM(1+4x,1-4*x)) where AGM denotes the arithmetic-geometric mean.

Original entry on oeis.org

1, 2, 0, 8, 2, 68, 32, 720, 464, 8480, 6656, 106368, 95912, 1390928, 1392512, 18734144, 20371650, 257955716, 300101760, 3613109008, 4448177412, 51302395528, 66289160512, 736588435360, 992578330048, 10674012880512, 14924667774976, 155890890782720, 225244659392784, 2291995151532576, 3410654921389824

Offset: 0

Joerg Arndt, Aug 14 2013

nonn

Convolution square is A092266.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A092266 (1+4*x)/AGM(1+4*x,1-4*x).

Cf. A081085 1/AGM(1,1-8*x), A053175 1/AGM(1,1-16*x), A090004 1/AGM(1,1-16*x)^(1/2), A089602 1/AGM(1,1-16*x)^(1/4).

CoefficientList[Series[Sqrt[2*(1 + 4*x)*EllipticK[1 - (1 + 4*x)^2/(1 - 4*x)^2] / (Pi*(1 - 4*x))], {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 27 2019 *)

Vec( 1/agm(1,(1-4*x)/(1+4*x)+O(x^66))^(1/2) ) \\ Joerg Arndt, Aug 14 2013

a(n) ~ 2^(2*n - 1/2) / (n*sqrt(Pi*log(n))) * (1 - (gamma + 3*log(2)) / (2*log(n)) + (3*gamma^2/8 + 9*gamma*log(2)/4 + 27*log(2)^2/8 - 1/16*Pi^2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

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A053175 Catalan-Larcombe-French sequence.

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A089603 G.f.: sqrt(1/agm(1, 1-8*x)) = sqrt(o.g.f. for A081085).

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A090004 Expansion of L(x)^(1/2), where L(x) is the g.f. for the Catalan Larcombe-French sequence A053175.

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A323385 Expansion of AGM(1,1-16*x) (where AGM denotes the arithmetic-geometric mean).

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A228156 Expansion of sqrt((1+4*x)/AGM(1+4*x,1-4*x)) where AGM denotes the arithmetic-geometric mean.

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A228156 Expansion of sqrt((1+4x)/AGM(1+4x,1-4*x)) where AGM denotes the arithmetic-geometric mean.