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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

G. C. Greubel, Table of n, a(n) for n = 0..223

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: A010789 A224863 A263075 * A246970 A246969 A134646

Adjacent sequences:  A217763 A217764 A217765 * A217767 A217768 A217769

KEYWORD

nonn,frac

AUTHOR

Juan Arias-de-Reyna, Mar 24 2013

STATUS

approved

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Last modified May 26 15:29 EDT 2019. Contains 323597 sequences. (Running on oeis4.)