A217766 Numerators for a rational approximation to Euler constant.

0, 2, 31, 1209, 87510, 10062642, 1676297196, 380613039300, 112785012934704, 42220061283665808, 19466179705605460320, 10832183496342326864160, 7154687325911822697398400, 5531732531984974533825018240, 4947671342477051367102277159680, 5067624845854754327998998304876800

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: A349071 A224863 A263075 * A246970 A246969 A358567

Adjacent sequences: A217763 A217764 A217765 * A217767 A217768 A217769

Keyword

nonn,frac

Author

Juan Arias-de-Reyna, Mar 24 2013

Status

approved