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A005258 Apery numbers: a(n) = sum_{k=0..n} C(n,k)^2 * C(n+k,k).
(Formerly M3057)
24
1, 3, 19, 147, 1251, 11253, 104959, 1004307, 9793891, 96918753, 970336269, 9807518757, 99912156111, 1024622952993, 10567623342519, 109527728400147, 1140076177397091, 11911997404064793, 124879633548031009, 1313106114867738897, 13844511065506477501 (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

Equals the main diagonal of square array A108625. - Paul D. Hanna, Jun 14 2005

This sequence is t_5 in Cooper's paper. - Jason Kimberley, Nov 25 2012

Conjecture: For each n=1,2,3,... the polynomial a_n(x) = sum_{k=0}^n C(n,k)^2*C(n+k,k)*x^k is irreducible over the field of rational numbers.- Zhi-Wei Sun, Mar 21 2013

REFERENCES

Roger 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.

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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).

LINKS

Simon Plouffe, Table of n, a(n) for n = 0..954

Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, May 24 1982, page 657.

F. Beukers, Another congruence for the Apéry numbers J. Number Theory 25 (1987), no. 2, 201-210.

S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).

E. Delaygue, Arithmetic properties of Apery-like numbers, arXiv preprint arXiv:1310.4131, 2013.

E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.

C. Elsner, On recurrence formulae for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.

Lalit Jain and Pavlos Tzermias, Beukers' integrals and Apery's recurrences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.1.

V. Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012.

R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820, 2014.

Simon Plouffe, The first 2553 Apery numbers

E. Rowland, R. Yassawi, Automatic congruences for diagonals of rational functions, arXiv preprint arXiv:1310.8635, 2013.

V. Strehl, Recurrences and Legendre transform

A. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3). An informal report. Math. Intelligencer 1 (1978/79), no 4, 195-203.

Eric Weisstein's World of Mathematics, Apery Number.

W. Zudilin, Approximations to -, di and trilogarithms, arXiv: math.NT/0409023, 2004-2005.

FORMULA

a(n) = hypergeom([n+1, -n, -n], [1, 1], 1). - Vladeta Jovovic, Apr 24 2003

(n+1)^2 * a(n+1) = (11*n^2+11*n+3) * a(n) + n^2 * a(n-1). - Matthijs Coster, Apr 28 2004

Let b(n) be the solution to the above recurrence with b(0) = 0, b(1) = 5. Then the b(n) are rational numbers with b(n)/a(n) -> zeta(2) very rapidly. The identity b(n)*a(n-1) - b(n-1)*a(n) = (-1)^(n-1)*5/n^2 leads to a series acceleration formula: zeta(2) = 5 * sum {n = 1..inf} 1/(n^2*a(n)*a(n-1)) = 5*[1/(1*3) + 1/(2^2*3*19) + 1/(3^2*19*147) + ...]. Similar results hold for the constant e: see A143413. - Peter Bala, Aug 14 2008

G.f.: hypergeom([1/12, 5/12],[1], 1728*x^5*(1-11*x-x^2)/(1-12*x+14*x^2+12*x^3+x^4)^3) / (1-12*x+14*x^2+12*x^3+x^4)^(1/4). - Mark van Hoeij, Oct 25 2011

a(n) ~ ((11+5*sqrt(5))/2)^(n+1/2)/(2*Pi*5^(1/4)*n). - Vaclav Kotesovec, Oct 05 2012

1/pi = 5 sqrt(47)/7614 Sum {n=0..infty} (-1)^n a(n)C(2n,n)(682n+71)/15228^n. [Cooper, equation (4)]. - Jason Kimberley, Nov 26 2012

a(-1 - n) = (-1)^n * a(n) if n>=0. a(-1 - n) = -(-1)^n * a(n) if n<0. - Michael Somos, Sep 18 2013

EXAMPLE

G.f. = 1 + 3*x + 19*x^2 + 147*x^3 + 1251*x^4 + 11253*x^5 + 104959*x^6 + ...

MAPLE

with (combinat): seq(add((multinomial(n+k, n-k, k, k))*binomial(n, k), k=0..n), n=0..18); # Zerinvary Lajos, Oct 18 2006

MATHEMATICA

a[n_] := HypergeometricPFQ[ {n+1, -n, -n}, {1, 1}, 1]; Table[ a[n], {n, 0, 18}] (* Jean-François Alcover, Jan 20 2012, after Vladeta Jovovic *)

PROG

(Haskell)

a005258 n = sum [a007318 n k ^ 2 * a007318 (n + k) k | k <- [0..n]]

-- Reinhard Zumkeller, Jan 04 2013

{a(n) = if( n<0, -(-1)^n * a(-1-n), sum(k=0, n, binomial(n, k)^2 * binomial(n+k, k)))} /* Michael Somos, Sep 18 2013 */

CROSSREFS

Cf. A002736, A005258, A005259, A005429, A005430, A108625, A143413, A218690, A218692.

Cf. A007318.

Sequence in context: A095002 A080833 A073516 * A131551 A074546 A054316

Adjacent sequences:  A005255 A005256 A005257 * A005259 A005260 A005261

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004

STATUS

approved

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Last modified May 26 23:22 EDT 2015. Contains 257849 sequences.