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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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57
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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, 279985580271435584, 2137790149251471680
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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Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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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FORMULA
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G.f.: 1 / AGM(1, 1 - 8*x).
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
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
a(n) = binomial(2*n, n)*hypergeom([1/2, -n, -n], [1, 1/2 - n], -1). - Peter Luschny, Apr 05 2022
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)
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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];
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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 */
(Python)
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
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CROSSREFS
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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.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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