A053175 -id:A053175 - cp's OEIS frontend

A081085 Expansion of 1 / AGM(1, 1 - 8*x) in powers of x.

1, 4, 20, 112, 676, 4304, 28496, 194240, 1353508, 9593104, 68906320, 500281280, 3664176400, 27033720640, 200683238720, 1497639994112, 11227634469668, 84509490017680, 638344820152784, 4836914483890112, 36753795855173776, 279985580271435584, 2137790149251471680

Offset: 0

Michael Somos, Mar 04 2003

AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre.
This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004
This is the exponential (also known as binomial) convolution of sequence A000984 (central binomial) with itself. See the V. Jovovic e.g.f. and a(n) formulas given below. - Wolfdieter Lang, Jan 13 2012
This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
The recursion (n+1)^2 * a(n+1) = (12*n^2+12*n+4) * a(n) - 32*n^2*a(n-1) with n=0 has zero coefficient for a(-1) and thus a(-1) is not determined uniquely by it, but defining a(-1) = 2^(-5/2) makes a(n) = a(-1-n) * 32^(n-1/2) true for all n in Z. - Michael Somos, Apr 05 2022

			G.f. = A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4304*x^5 + 28496*x^6 + ...

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

Seiichi Manyama, Table of n, a(n) for n = 0..1110 (terms 0..200 from Vincenzo Librandi)
B. Adamczewski, J. P. Bell, and E. Delaygue, Algebraic independence of G-functions and congruences "a la Lucas", arXiv preprint arXiv:1603.04187 [math.NT], 2016.
Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982, page 657.
Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023.
E. Delaygue, Arithmetic properties of Apery-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013-2015.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See E p. 2.
S. Herfurtner, Elliptic surfaces with four singular fibres, Mathematische Annalen, 1991. Preprint.
Xiao-Juan Ji and Zhi-Hong Sun, Congruences for Catalan-Larcombe-French numbers, arXiv:1505.00668 [math.NT], 2015.
Bradley Klee, Checking Weierstrass data, 2023.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.
Stéphane Ouvry and Alexios Polychronakos, Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers, arXiv:2006.06445 [math-ph], 2020.
Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020.
D. Zagier, Integral solutions of Apery-like recurrence equations. See line E in sporadic solutions table of page 5.

Cf. A053175, A089603, A091401.
The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

seq(simplify(binomial(2*n, n)*hypergeom([ -n, -n, 1/2], [1, -n+1/2], -1)), n = 0..22); # Peter Bala, Jul 25 2024

Table[Sum[Binomial[n,k]*Binomial[2*n-2*k,n-k]*Binomial[2*k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/2, 1/2, 1, 16 x (1 - 4 x)], {x, 0, n}]; (* Michael Somos, Oct 25 2014 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ 1 / NestWhile[ {(#[[1]] + #[[2]])/2, Sqrt[#[[1]] #[[2]]]} &, {1, Series[ 1 - 8 x, {x, 0, n}]}, #[[1]] =!= #[[2]] &] // First, {x, 0, n}]]; (* Michael Somos, Oct 27 2014 *)
CoefficientList[Series[2*EllipticK[1/(1 - 1/(4*x))^2] / (Pi*(1 - 4*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 13 2019 *)
a[n_] := Binomial[2 n, n] HypergeometricPFQ[{1/2, -n, -n},{1, 1/2 - n}, -1];
Table[a[n], {n, 0, 20}] (* Peter Luschny, Apr 05 2022 *)

{a(n) = if( n<0, 0, polcoeff( 1 / agm( 1, 1 - 8 * x + x * O(x^n)), n))};

{a(n) = if( n<0,0, 4^n * sum( k=0, n\2, binomial( n, 2*k) * binomial( 2*k, k)^2 / 16^k))};

{a(n)=n!*polcoeff(sum(k=0,n,(2*k)!*x^k/(k!)^3 +x*O(x^n))^2,n)} /* Paul D. Hanna, Sep 04 2009 */

from math import comb
def A081085(n): return sum((1<<(n-(m:=k<<1)<<1))*comb(n,m)*comb(m,k)**2 for k in range((n>>1)+1)) # Chai Wah Wu, Jul 09 2023

G.f.: 1 / AGM(1, 1 - 8*x).
E.g.f.: exp(4*x)*BesselI(0, 2*x)^2. - Vladeta Jovovic, Aug 20 2003
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*n-2*k, n-k)*binomial(2*k, k) = binomial(2*n, n)*hypergeom([ -n, -n, 1/2], [1, -n+1/2], -1). - Vladeta Jovovic, Sep 16 2003
D-finite with recurrence (n+1)^2 * a(n+1) = (12*n^2+12*n+4) * a(n) - 32*n^2*a(n-1). - Matthijs Coster, Apr 28 2004
E.g.f.: [Sum_{n>=0} binomial(2n,n)*x^n/n! ]^2. - Paul D. Hanna, Sep 04 2009
G.f.: Gaussian Hypergeometric function 2F1(1/2, 1/2; 1; 16*x-64*x^2). - Mark van Hoeij, Oct 24 2011
a(n) = 2^(-n) * A053175(n).
a(n) ~ 2^(3*n+1)/(Pi*n). - Vaclav Kotesovec, Oct 13 2012
0 = x*(x+4)*(x+8)*y'' + (3*x^2 + 24*x + 32)*y' + (x+4)*y, where y(x) = A(x/-32). - Gheorghe Coserea, Aug 26 2016
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k)^2. - Seiichi Manyama, Apr 02 2017
a(n) = (1/Pi)^2*Integral_{0 <= x, y <= Pi} (4*cos(x)^2 + 4*cos(y)^2)^n dx dy. - Peter Bala, Feb 10 2022
a(n) = a(-1-n)*32^(n-1/2) and 0 = +a(n)*(+a(n+1)*(+32768*a(n+2) -23552*a(n+3) +3072*a(n+4)) +a(n+2)*(-8192*a(n+2) +8448*a(n+3) -1248*a(n+4)) +a(n+3)*(-512*a(n+3) +96*a(n+4))) +a(n+1)*(+a(n+1)*(-5120*a(n+2) +3840*a(n+3) -512*a(n+4)) +a(n+2)*(+1536*a(n+2) -1728*a(n+3) +264*a(n+4)) +a(n+3)*(+120*a(n+3) -23*a(n+4))) +a(n+2)*(+a(n+2)*(-32*a(n+2) +48*a(n+3) -8*a(n+4)) +a(n+3)*(-5*a(n+3) +a(n+4))) for all n in Z. - Michael Somos, Apr 04 2022
From Bradley Klee, Jun 05 2023: (Start)
The g.f. T(x) obeys a period-annihilating ODE:
0=4*(-1 + 8*x)*T(x) + (1 - 24*x + 96*x^2)*T'(x) + x*(-1 + 4*x)*(-1 + 8*x)*T''(x).
The periods ODE can be derived from the following Weierstrass data:
g2 = 3*(1 - 16*(1 - 8*x)^2 + 16*(1 - 8*x)^4);
g3 = 1 + 30*(1 - 8*x)^2 - 96*(1 - 8*x)^4 + 64*(1 - 8*x)^6;
which determine an elliptic surface with four singular fibers. (End)
G.f.: Sum_{n>=0} binomial(2*n,n)^2 * x^n * (1 - 4*x)^n. - Paul D. Hanna, Apr 18 2024
From Peter Bala, Jul 25 2024: (Start)
a(n) = 2*Sum_{k = 1..n} (k/n)*binomial(n, k)*binomial(2*n-2*k, n-k)*binomial(2*k, k) for n >= 1.
a(n-1) = (1/2)*Sum_{k = 1..n} (k/n)^2*binomial(n, k)*binomial(2*n-2*k, n-k)*  binomial(2*k, k) for n >= 1. Cf. A002895. (End)

A014330 Exponential convolution of Catalan numbers with themselves.

Original entry on oeis.org

1, 2, 6, 22, 92, 424, 2108, 11134, 61748, 356296, 2123720, 13002840, 81417520, 519550880, 3369559864, 22161337742, 147544048324, 992923683912, 6746101933304, 46226667046360, 319199694771696, 2219445498261152, 15529758665102416, 109291258152550712

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A014333, A053175, A126869, A138364.

A014330:= func< n | (&+[Binomial(n,k)*Catalan(k)*Catalan(n-k): k in [0..n]]) >;
[A014330(n): n in [0..40]]; // G. C. Greubel, Jan 06 2023

Table[Sum[Binomial[n, k]*Binomial[2*k, k]/(k+1)*Binomial[2*n-2*k, n-k]/(n-k+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Feb 25 2014 *)

A014330(n)=sum(k=0,n,binomial(n,k)*A000108(k)*A000108(n-k))  \\ M. F. Hasler, Jan 13 2012

def c(n): return catalan_number(n)
def A014330(n): return sum( binomial(n,k)*c(k)*c(n-k) for k in range(n+1))
[A014330(n) for n in range(41)] # G. C. Greubel, Jan 06 2023

From Vladeta Jovovic, Jan 01 2004: (Start)
E.g.f.: exp(4*x)*(BesselI(0, 2*x) - BesselI(1, 2*x))^2.
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*k, k)/(k+1)*binomial(2*n-2*k, n-k)/(n-k+1).
a(n) = 4^n*Sum_{k=0..n} (-4)^(-k)*binomial(n, k)*binomial(k, floor(k/2))*binomial(k+1, floor((k+1)/2)).
a(n) = binomial(2*n, n)/(n+1)*hypergeometric3F2([-n-1, -n, 1/2], [2, 1/2-n], -1). (End)
(n + 1)*(n + 2)*a(n) = 4*(3*n^2 + n - 1)*a(n - 1) - 32*(n - 1)^2*a(n - 2). - Vladeta Jovovic, Jul 15 2004
a(n) = Sum_{k=0..n} binomial(n,k)*A000108(k)*A000108(n-k). - Philippe Deléham, Aug 23 2006
a(n) = (4*A053175(n) - A053175(n+1)/4) / ((n+2)*2^n). - Mark van Hoeij, Jul 02 2010
G.f.: (1-6*x)*hypergeometric2F1([1/2, 1/2],[2],16*x^2/(4*x-1)^2)/(2*x*(4*x-1)) - x*(8*x-1)*hypergeometric2F1([3/2, 3/2],[3],16*x^2/(4*x-1)^2)/(4*x-1)^3 + 1/(2*x). - Mark van Hoeij, Oct 25 2011
E.g.f.: hypergeometric1F1([1/2],[2],4*x)^2, coinciding with the above given e.g.f. - Wolfdieter Lang, Jan 13 2012
a(n) ~ 8^(n+1) / (Pi*n^3). - Vaclav Kotesovec, Feb 25 2014

More terms from Vincenzo Librandi, Feb 27 2014

A065409 Fennessey-Larcombe-French sequence.

Original entry on oeis.org

1, 8, 144, 2432, 40000, 649728, 10486784, 168681472, 2708038656, 43425996800, 695894425600, 11146676797440, 178493059563520, 2857665426882560, 45744737668300800, 732196083173687296, 11718755500209471488

Offset: 0

Peter J Larcombe, Nov 14 2001

Numbers appearing as coefficients in the series expansion of an elliptic integral of the second kind. Defining f(x; c) = [1 - c^2*sin^2(x)]^(1/2), consider the function E(c) obtained by integrating f(x; c) with respect to x between 0 and Pi/2. E(c) is transformed and written as a power series in c (through an intermediate variable) which acts as a generating function for the sequence.

E'(k) is complete elliptic integral of second kind evaluated at k'. - Michael Somos, Mar 04 2003

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, Power series identities generated by two recent integer sequences, Bulletin ICA, 43 (2005), 85-95.
P. J. Larcombe, A new asymptotic relation between two recent integer sequences, Congressus Numerantium, 175 (2005), 111-116.
P. J. Larcombe, D. R. French and E. J. Fennessey, The Fennessey-Larcombe-French sequence {1, 8, 144, 2432, 40000, ...}: formulation and asymptotic form, Congressus Numerantium, 158 (2002), 179-190.
P. J. Larcombe, D. R. French and E. J. Fennessey, The Fennessey-Larcombe-French sequence {1, 8, 144, 2432, 40000, ...}: a recursive formulation and prime factor decomposition, Congressus Numerantium, 160 (2003), 129-137.

T. D. Noe, Table of n, a(n) for n=0..200
Arthur L. B. Yang, James J. Y. Zhao, Log-concavity of the Fennessey-Larcombe-French Sequence, arXiv:1503.02151 [math.CO], 2015.

Cf. A053175, A010370.

a[n_] := 8^n*HypergeometricPFQ[{1/2, 5/4, 1/2-n/2, -n/2}, {1/4, 1, 1}, 1 ]; Table[ a[n], {n, 0, 16}] (* Jean-François Alcover, Jan 31 2012, from first formula *)
Table[2^n Sum[(4k^2-2k-1)/(2k-1) Binomial[n,k]Binomial[2n-2k,n-k] Binomial[ 2k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Mar 18 2012 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ Binomial[2 k, k]^2 / (1 - 2 k) (2 x - 16 x^2)^k, {k, 0, n}] / (1 - 16 x), {x, 0, n}]]; (*Michael Somos, Jul 10 2017 *)
a[ n_] := If [n < 0, 0, n! SeriesCoefficient[ Exp[8 x] BesselI[0, 4 x] (BesselI[0, 4 x] + 16 x BesselI[1, 4 x]), {x, 0, n}]]; (* Michael Somos, Jul 10 2017 *)

{a(n) = my(A); if( n<0, 0, A =a gm(1, 1 - 16*x + x*O(x^n)); polcoeff((1 - 16*x - 2*x*(1 - 8*x) * log(A)') / A, n))}; /* Michael Somos, Mar 04 2003 */

{a(n) = if( n<0, 0, polcoeff( sum(k=0, n, binomial(2*k,k)^2 / (1 - 2*k) * (2*x - 16*x^2)^k, x*O(x^n)) / (1 - 16*x), n))}; /* Michael Somos, Mar 04 2003 */

a(n) = 8^n * 4F3( [5/4, 1/2, (1/2)-n/2, -n/2], [1, 1, 1/4] | 1 ).
G.f.: F(-1/2, 1/2; 1; 32*x - 256*x^2) / (1 - 16*x) = E'(1 - 16*x) / (Pi/2 * (1 - 16*x)). - Michael Somos, Mar 04 2003
a(n)*(n^3 - n^2) = a(n-1)*(8 - 32*n^2 + 24*n^3) + a(n-2)*(256*n^2 - 128*n^3). - Michael Somos, Mar 04 2003
a(n) = 2^n*Sum_{k=0..n} (4*k^2-2*k-1)/(2*k-1)*binomial(n, k)*binomial(2*n-2*k, n-k)*binomial(2*k, k). - Vladeta Jovovic, Jun 02 2005
E.g.f.: exp(8*x)*BesselI(0, 4*x)*(BesselI(0, 4*x)+16*x*BesselI(1, 4*x)). - Vladeta Jovovic, Jun 02 2005
a(n) = (n+1)^2*(A053175(n+1)-8*A053175(n))/(8*n) for n>0. - Mark van Hoeij, Oct 31 2011
a(n) ~ 2^(4*n+1)/Pi. - Vaclav Kotesovec, Aug 13 2013

A228289 Determinant of the p_n X p_n matrix with (i,j)-entry equal to D(i+j) for all i,j = 0,...,p_n-1, where D(k) = A002895(k) is the k-th Domb number and p_n is the n-th prime.

Original entry on oeis.org

12, 2448, 428587718400, 4994319435309277891448832, 191901511752240055024005979549622856313555581586068578283027431424, 637213222716753775758429677219909335140503764595701930312765250413280716374852064945052319744

Offset: 1

Zhi-Wei Sun, Aug 19 2013

nonn

Conjecture: If p_n == 1 (mod 3) and p_n = x^2 + 3*y^2 with x and y integers, then we have a(n) == (-1)^{(p_n-1)/2}*(4*x^2-2*p_n) (mod p_n^2). In the case p_n == 2 (mod 3), we have a(n) == 0 (mod p_n^2).
Zhi-Wei Sun also made the following similar conjecture:
If p is an odd prime and b(p) is the p X p determinant with (i,j)-entry equal to A053175(i+j) for all i,j = 0,...,p-1, then we have the congruence b(p) == (-1)^{(p-1)/2} (mod p^2).

Zhi-Wei Sun, Conjectures and results on x^2 mod p^2 with 4*p = x^2 + d*y^2, in: Number Theory and Related Area (eds., Y. Ouyang, C. Xing, F. Xu and P. Zhang), Higher Education Press & International Press, Beijing and Boston, 2013, pp. 147-195.

Cf. A002895, A053175, A225776, A228143.

d[n_]:=Sum[Binomial[n,k]^2*Binomial[2k,k]Binomial[2(n-k),n-k],{k,0,n}]
a[n_]:=Det[Table[d[i+j],{i,0,Prime[n]-1},{j,0,Prime[n]-1}]]
Table[a[n],{n,1,8}]

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

cp's OEIS Frontend

A089602 Expansion of L(x)^(1/4), where L(x) = o.g.f. for A053175.

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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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A081085 Expansion of 1 / AGM(1, 1 - 8*x) in powers of x.

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A014330 Exponential convolution of Catalan numbers with themselves.

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A065409 Fennessey-Larcombe-French sequence.

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A228289 Determinant of the p_n X p_n matrix with (i,j)-entry equal to D(i+j) for all i,j = 0,...,p_n-1, where D(k) = A002895(k) is the k-th Domb number and p_n is the n-th prime.

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