A038533 Denominator of coefficients of both EllipticK/Pi and EllipticE/Pi.

2, 8, 128, 512, 32768, 131072, 2097152, 8388608, 2147483648, 8589934592, 137438953472, 549755813888, 35184372088832, 140737488355328, 2251799813685248, 9007199254740992, 9223372036854775808, 36893488147419103232, 590295810358705651712, 2361183241434822606848

Offset: 0

Author

Wouter Meeussen, revised Jan 03 2001

Keywords

• nonn
• frac

Comments

Denominators are powers of 2 since EllipticK(x) = Pi * Sum_{n>=0} 2^(-4*n-1) * binomial(2*n,n)^2 * x^n and EllipticE(x) = Pi * Sum_{n>=0} 2^(-4*n-1) (-1)^(2*n) * binomial(2*n,n)^2 /(-2*n+1) * x^n.

Links

• David P. Roberts and Fernando Rodriguez Villegas, Hypergeometric Motives, Notices of the American Mathematical Society, Vol. 69, No. 6 (2022), pp. 914-929; arXiv preprint, arXiv:2109.00027 [math.AG], 2021. See eq. (1.2), p. 914.
• Index to divisibility sequences.

Crossrefs

Cf. A000120, A038534, A038535.

Equals 2*A056982(n).

Programs

Mathematica

a[n_] := 2^(4*n - 2*DigitCount[n, 2, 1] + 1); Array[a, 20, 0] (* Amiram Eldar, Aug 03 2023 *)

PARI

a(n)=my(s=n); while(n>>=1, s+=n); 2<<(2*s) \\ Charles R Greathouse IV, Apr 07 2012

Formula

a(n) = 2^(1+4*n-2*w(n)) with w(n) = A000120(n) = number of 1's in binary expansion of n.