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 A036917 a(n) = (16*(n-1/2)*(2*n^2-2*n+1)*a(n-1)-256*(n-1)^3*a(n-2))/n^3. 41
 1, 8, 88, 1088, 14296, 195008, 2728384, 38879744, 561787864, 8206324928, 120929313088, 1794924383744, 26802975999424, 402298219288064, 6064992788397568, 91786654611673088, 1393772628452578264, 21227503080738294464, 324160111169327247424 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES M. Petkovsek et al., "A=B", Peters, p. ix of second printing. LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..500 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. E. Delaygue, Arithmetic properties of Apery-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013. Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11. Ji-Cai Liu and He-Xia Ni, Supercongruences for Almkvist--Zudilin sequences, arXiv:2004.07652 [math.NT], 2020. See Vn. N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98). Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020. See Vn. Zhi-Hong Sun, Congruences for two types of Apery-like sequences, arXiv:2005.02081 [math.NT], 2020. FORMULA a(n) = Sum_{k=0..n} (C(2 * (n-k), n-k) * C(2 * k, k))^2. [corrected by Tito Piezas III, Oct 19 2010] G.f.: (4/Pi^2)*EllipticK(4*x^(1/2))^2. - Vladeta Jovovic, Dec 01 2003 a(n) = hypergeom([1/2, 1/2, -n, -n], [1, 1/2-n, 1/2-n], 1) * 4^n * (2n-1)!!^2 / n!^2. - Vladimir Reshetnikov, Mar 08 2014 a(n) ~ 2^(4*n+1) * log(n) / (n*Pi^2) * (1 + (4*log(2) + gamma)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Nov 28 2015 G.f. y=A(x) satisfies: 0 = x^2*(16*x - 1)^2*y''' + 3*x*(16*x - 1)*(32*x - 1)*y'' + (1792*x^2 - 112*x + 1)*y' + 8*(32*x - 1)*y. - Gheorghe Coserea, Jul 03 2018 G.f.: 1 / AGM(1, sqrt(1 - 16*x))^2. - Vaclav Kotesovec, Oct 01 2019 It appears that a(n) is equal to the coefficient of (x*y*z*t)^n in the expansion of (1+x+y+z-t)^n * (1+x+y-z+t)^n * (1+x-y+z+t)^n * (1-x+y+z+t)^n. Cf. A000172. - Peter Bala, Sep 21 2021 G.f. y = A(x) satisfies 0 = x*(1 - 16*x)*(2*y''*y - y'*y') + 2*(1 - 32*x)*y*y' - 16*y*y. - Michael Somos, May 29 2023 Expansion of theta_3(0, q)^4 in powers of m/16 where the modulus m = k^2. - Michael Somos, May 30 2023 EXAMPLE G.f. = 1 + 8*x + 88*x^2 + 1088*x^3 + 14296*x^5 + 195008*x^5 + ... - Michael Somos, May 29 2023 MATHEMATICA a[n_] := (16 (n - 1/2)(2*n^2 - 2*n + 1)a[n - 1] - 256(n - 1)^3 a[n - 2])/n^3; a[0] = 1; a[1] = 8; Array[a, 19, 0] (* Or *) f[n_] := Sum[(Binomial[2 (n - k), n - k] Binomial[2 k, k])^2, {k, 0, n}]; Array[f, 19, 0] (* Or *) lmt = 20; Take[ 4^Range[0, 2 lmt]*CoefficientList[ Series[(4/Pi^2) EllipticK[4 x^(1/2)]^2, {x, 0, lmt}], x^(1/2)], lmt] (* Robert G. Wilson v *) a[n_] := HypergeometricPFQ[{1/2, 1/2, -n, -n}, {1, 1/2-n, 1/2-n}, 1] * 4^n * (2n-1)!!^2 / n!^2 (* Vladimir Reshetnikov, Mar 08 2014 *) a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, EllipticNomeQ[16*x]]^4, {x, 0, n}]; (* Michael Somos, May 30 2023 *) PROG (Haskell) a036917 n = sum \$ map (\k -> (a007318 (2*n-2*k) (n-k))^2 * (a007318 (2*k) k)^2) [0..n] -- Reinhard Zumkeller, May 24 2012 (PARI) for(n=0, 25, print1(sum(k=0, n, (binomial(2*n-2*k, n-k) *binomial(2*k, k))^2), ", ")) \\ G. C. Greubel, Oct 24 2017 (PARI) {a(n) = if(n<0, 0, polcoeff(agm(1, sqrt(1 - 16*x + x*O(x^n)))^-2, n)}); /* Michael Somos, May 29 2023 */ CROSSREFS Cf. A002894, A036915, A057703. Cf. A007318, A036916, A036829. 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.) Sequence in context: A250166 A247738 A115864 * A003497 A051605 A271268 Adjacent sequences: A036914 A036915 A036916 * A036918 A036919 A036920 KEYWORD nonn,easy AUTHOR N. J. A. Sloane STATUS approved

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Last modified December 2 06:18 EST 2023. Contains 367509 sequences. (Running on oeis4.)