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A217766 Numerators for a rational approximation to Euler constant. 2
0, 2, 31, 1209, 87510, 10062642, 1676297196, 380613039300, 112785012934704, 42220061283665808, 19466179705605460320, 10832183496342326864160, 7154687325911822697398400, 5531732531984974533825018240, 4947671342477051367102277159680, 5067624845854754327998998304876800 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
a(n)/A217767(n) converges to Euler's constant.
0 < A217766(n)/A217767(n)-gamma < 2*Pi*exp(-2*sqrt(2n))(1+O(n^(-1/2))).
REFERENCES
A. I. Aptekarev (Editor), Rational approximants for Euler's constant and recurrence relations, Collected papers, Sovrem. Probl. Mat. ("Current Problems in Mathematics") Vol. 9, MIAN (Steklov Institute), Moscow (2007), 84pp (Russian)
LINKS
Kh. Hessami Pilehrood, T. Hessami Pilehrood, On a continued fraction expansion for Euler's constant, Journal of Number Theory, 133 (2013) 769--786.
D. N. Tulyakov, A system of recurrence relations for rational approximations of the Euler constant, (Russian) Mat. Zametki 85 (2009), No. 5 , 782-787. Translation: Mathematical Notes 85 (2009), No. 5, 746-750.
FORMULA
a(n) = Sum_{k=0..n} binomial(n,k)^2 *(n+k)!*(H(n+k)+2*H(n-k)-2*H(k)) where H(n) is the n-th harmonic number. (Pilehrood)
(16*n - 15)*a(n+1) = (128*n^3 + 40*n^2 - 82*n - 45)*a(n) - n^2*(256*n^3 -240*n^2 +64*n-7)*a(n-1) +(16*n + 1)*n^2*(n - 1)^2*a(n-2), (the integrality has been proved by Tulyakov).
EXAMPLE
G.f. = 2*x + 31*x^2 + 1209*x^3 + 87510*x^4 + 10062642*x^5 + ...
MATHEMATICA
Table[ Sum[ Binomial[n, k]^2 (n + k)! (HarmonicNumber[n + k] + 2 HarmonicNumber[n - k] - 2 HarmonicNumber[k]), {k, 0, n}], {n, 0, 20}]
PROG
(PARI) {a(n) = my(H = k->sum(i=1, k, 1/i)); sum(k=0, n, binomial(n, k)^2 * (n+k)! * (H(n+k) + 2*H(n-k) - 2*H(k)))}; /* Michael Somos, Nov 25 2016 */
CROSSREFS
Cf. A217767 (denominators).
Sequence in context: A349071 A224863 A263075 * A246970 A246969 A358567
KEYWORD
nonn,frac
AUTHOR
Juan Arias-de-Reyna, Mar 24 2013
STATUS
approved

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Last modified February 23 18:28 EST 2024. Contains 370283 sequences. (Running on oeis4.)