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 A006077 (n+1)^2*a(n+1) = (9n^2+9n+3)*a(n) - 27*n^2*a(n-1), with a(0) = 1 and a(1) = 3. (Formerly M2775) 51
 1, 3, 9, 21, 9, -297, -2421, -12933, -52407, -145293, -35091, 2954097, 25228971, 142080669, 602217261, 1724917221, 283305033, -38852066421, -337425235479, -1938308236731, -8364863310291, -24286959061533, -3011589296289, 574023003011199, 5028616107443691 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004 Conjecture: Let W(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. If n == 1 (mod 3) then W(n) = 0. When n == 0 or 2 (mod 3), W(n)*(-1)^(floor((n+1)/3))/6^n is always a positive odd integer. - Zhi-Wei Sun, Aug 21 2013 Conjecture: Let p == 1 (mod 3) be a prime, and write 4*p = x^2 + 27*y^2 with x, y integers and x == 1 (mod 3). Then W(p-1) == (-1)^{(p+1)/2}*(x-p/x) (mod p^2), where W(n) is defined as the above. - Zhi-Wei Sun, Aug 23 2013 This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017 Diagonal of rational functions 1/(1 - (x^2*y + y^2*z - z^2*x + 3*x*y*z)), 1/(1 - (x^3 + y^3 - z^3 + 3*x*y*z)), 1/(1 + x^3 + y^3 + z^3 - 3*x*y*z). - Gheorghe Coserea, Aug 04 2018 REFERENCES Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1400 (terms 0..200 from T. D. Noe) Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982. A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015. M. Coster, Email, Nov 1990 Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See B p. 2. 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-Wei Sun, Three mysterious conjectures on Hankel-type determinants, a message to Number Theory List, August 22, 2013. Zhi-Wei Sun, Connections between p = x^2+3*y^2 and Franel numbers, J. Number Theory 133(2013), 2914-2928. FORMULA G.f.: hypergeom([1/3, 2/3], , x^3/(x-1/3)^3) / (1-3*x). - Mark van Hoeij, Oct 25 2011 a(n) = Sum_{k=0..floor(n/3)}(-1)^k*3^(n-3k)*C(n,3k)*C(2k,k)*C(3k,k). - Zhi-Wei Sun, Aug 21 2013 0 = x*(x^2+9*x+27)*y'' + (3*x^2 + 18*x + 27)*y' + (x + 3)*y, where y(x) = A(x/-27). - Gheorghe Coserea, Aug 26 2016 a(n) = 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], 1). - Peter Luschny, Nov 01 2017 EXAMPLE G.f. = 1 + 3*x + 9*x^2 + 21*x^3 + 9*x^4 - 297*x^5 - 2421*x^6 - 12933*x^7 - ... MAPLE a := n -> 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], 1): seq(simplify(a(n)), n=0..24); # Peter Luschny, Nov 01 2017 MATHEMATICA Table[Sum[(-1)^k*3^(n - 3*k)*Binomial[n, 3*k]*Binomial[2*k, k]* Binomial[3*k, k], {k, 0, Floor[n/3]}], {n, 0, 50}] (* G. C. Greubel, Oct 24 2017 *) a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/3, 2/3}, {1}, x^3 / (x - 1/3)^3 ] / (1 - 3 x), {x, 0, n}]; (* Michael Somos, Nov 01 2017 *) PROG (PARI) subst(eta(q)^3/eta(q^3), q, serreverse(eta(q^9)^3/eta(q)^3*q)) \\ (generating function) Helena Verrill (verrill(AT)math.lsu.edu), Apr 20 2009 [for (-1)^n*a(n)] (PARI) diag(expr, N=22, var=variables(expr)) = {   my(a = vector(N));   for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));   for (n = 1, N, a[n] = expr;     for (k = 1, #var, a[n] = polcoeff(a[n], n-1)));   return(a); }; diag(1/(1 + x^3 + y^3 + z^3 - 3*x*y*z), 25) (PARI) seq(N) = {   my(a = vector(N)); a = 3; a = 9;   for (n = 2, N-1, a[n+1] = ((9*n^2+9*n+3)*a[n] - 27*n^2*a[n-1])/(n+1)^2);   concat(1, a); }; seq(24) \\ test: y=subst(Ser(seq(202)), 'x, -'x/27); 0 == x*(x^2+9*x+27)*y'' + (3*x^2+18*x+27)*y' + (x+3)*y \\ Gheorghe Coserea, Nov 09 2017 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); (-1)^n * polcoeff( subst(eta(x + A)^3 / eta(x^3 + A), x, serreverse( x * eta(x^9 + A)^3 / eta(x + A)^3)), n))}; /* Michael Somos, Nov 01 2017 */ CROSSREFS Related to diagonal of rational functions: A268545-A268555. Cf. 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: A146219 A197403 A305618 * A291898 A330987 A109612 Adjacent sequences:  A006074 A006075 A006076 * A006078 A006079 A006080 KEYWORD sign AUTHOR EXTENSIONS More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000 STATUS approved

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Last modified September 24 16:19 EDT 2022. Contains 356943 sequences. (Running on oeis4.)