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A005259
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Apery (Ap\'{e}ry) numbers: Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2.
(Formerly M4020)
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24
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1, 5, 73, 1445, 33001, 819005, 21460825, 584307365, 16367912425, 468690849005, 13657436403073, 403676083788125, 12073365010564729, 364713572395983725
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Prime Apery numbers include a(1) = 5, a(2) = 73, a(12) = 12073365010564729 and a(24). Semiprime central Delannoy numbers include a(4) = 33001 = 61 * 541. - Jonathan Vos Post (jvospost3(AT)gmail.com), May 22 2005
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REFERENCES
| R. Apery, Irrationalite de zeta(2) et zeta(3), in Journees Arith. de Luminy. Colloque International du Centre National de la Recherche Scientifique (CNRS) held at the Centre Universitaire de Luminy, Luminy, Jun 20-24, 1978. Asterisque, 61 (1979), 11-13.
R. Apery, ``Interpolation de fractions continues et irrationalit\'{e} de certaines constantes,'' in Math\'{e}matiques, Minist\`{e}re universit\'{e}s (France), Comit\'{e} travaux historiques et scientifiques. Bull. Section Sciences, Vol. 3, pp. 243-246, 1981.
F. Beukers, Another congruence for the Apery numbers. J. Number Theory 25 (1987), no. 2, 201-210.
F. Beukers, Consequences of Apery's work on zeta(3), in "Zeta(3) irrationnel: les retombees", Rencontres Arithmetiques de Caen, June 2-3, 1995 [Mentions divisibility of a(n) by powers of 5 and powers of 11]
C. Elsner, On recurrence formulae for sums involving binomial coefficients, Fib. Q., 43 (No. 1, 2005), 31-45.
W. Koepf, Hypergeometric Summation, Vieweg, 1998, p. 146.
M. Kontsevich and D. Zagier, Periods, pp. 771-808 of B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Volker Strehl, Binomial identities -- combinatorial and algorithmic aspects, Discrete Mathematics, Vol. 136 (1994), 309-346. [From David Callan (callan(AT)stat.wisc.edu), Aug 27 2009]
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
J.-P. Allouche, A remark on Apery's numbers, J. Comput. Appl. Math. 83 (1997), 123-125.
E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.
S. Fischler, Irrationalite de valeurs de zeta
L. Lipshitz and A. J. van der Poorten, Rational functions, diagonals, automata and arithmetic
V. Strehl, Recurrences and Legendre transform
Eric Weisstein's World of Mathematics, Apery Number
Eric Weisstein's World of Mathematics, Strehl Identities
Eric Weisstein's World of Mathematics, Schmidt's Problem
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FORMULA
| (n+1)^3*a(n+1) = (34*n^3 + 51*n^2 + 27*n +5)*a(n) - n^3*a(n-1), n >= 1.
Representation as a special value of the hypergeometric function 4F3, in Maple notation: a(n)=hypergeom([n+1, n+1, -n, -n], [1, 1, 1], 1), n=0, 1... - Karol A. Penson (penson(AT)lptl.jussieu.fr) Jul 24 2002
a(n) = Sum( k>=0, A063007(n, k)*A000172(k)). A000172 = Franel numbers. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Aug 14 2003
G.f.: (-1/2)*(3*x-3+(x^2-34*x+1)^(1/2))*(x+1)^(-2)*hypergeom([1/3,2/3],[1],(-1/2)*(x^2-7*x+1)*(x+1)^(-3)*(x^2-34*x+1)^(1/2)+(1/2)*(x^3+30*x^2-24*x+1)*(x+1)^(-3))^2 - Mark van Hoeij, Oct 29 2011.
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MAPLE
| a := proc(n) option remember; if n=0 then 1 elif n=1 then 5 else (n^(-3))* ( (34*(n-1)^3 + 51*(n-1)^2 + 27*(n-1) +5)*a((n-1)) - (n-1)^3*a((n-1)-1)); fi; end;
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MATHEMATICA
| Table[HypergeometricPFQ[{-n, -n, n+1, n+1}, {1, 1, 1}, 1], {n, 0, 13}] (* From Jean-François Alcover, Apr 1 2011 *)
Table[Sum[(Binomial[n, k]Binomial[n+k, k])^2, {k, 0, n}], {n, 0, 30}] (* From Harvey P. Dale, Oct 15 2011 *)
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CROSSREFS
| Cf. A002736, A005258, A005259, A005429, A005430, A059415, A059416.
Cf. A063007, A000172.
Sequence in context: A155662 A159509 A127167 * A195636 A062440 A197764
Adjacent sequences: A005256 A005257 A005258 * A005260 A005261 A005262
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Simon Plouffe (simon.plouffe(AT)gmail.com), N. J. A. Sloane (njas(AT)research.att.com).
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