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A081085
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Expansion of 1 / AGM(1, 1 - 8*x) in powers of x.
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56
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1, 4, 20, 112, 676, 4304, 28496, 194240, 1353508, 9593104, 68906320, 500281280, 3664176400, 27033720640, 200683238720, 1497639994112, 11227634469668, 84509490017680, 638344820152784, 4836914483890112, 36753795855173776
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
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LINKS
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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.
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.
Xiao-Juan Ji and Zhi-Hong Sun, Congruences for Catalan-Larcombe-French numbers, arXiv:1505.00668 [math.NT], 2015.
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.
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FORMULA
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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
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
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..[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
a(n) = binomial(2*n, n)*hypergeom([1/2, -n, -n], [1, 1/2 - n], -1). - Peter Luschny, Apr 05 2022
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EXAMPLE
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G.f. = A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4304*x^5 + 28496*x^6 + ...
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MATHEMATICA
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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 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / agm( 1, 1 - 8 * x + x * O(x^n)), n))};
(PARI) {a(n) = if( n<0, 0, 4^n * sum( k=0, n\2, binomial( n, 2*k) * binomial( 2*k, k)^2 / 16^k))};
(PARI) {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 */
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CROSSREFS
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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.
Sequence in context: A227726 A080609 A003645 * A212326 A192624 A209200
Adjacent sequences: A081082 A081083 A081084 * A081086 A081087 A081088
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KEYWORD
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nonn,easy
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AUTHOR
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Michael Somos, Mar 04 2003
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STATUS
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approved
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