A389637 Numerators of coefficients of 1- Pi/(2*EllipticK).

0, 1, 5, 11, 469, 1379, 17223, 56001, 11998869, 41064827, 571915951, 2018982161, 115338112823, 415720532641, 6041874952949, 22103950817043, 20825721430968213, 77047750289886219, 1145470055108455527, 4274935497276922857, 256206642255178772127

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

Numerators of the coefficients with even exponent of 1 - Pi/(2*EllipticK(x^2)) = 1-AGM(1-x, 1+x) = Sum_{k>=0} (a(k)/A056982(k))*x^(2*k) with 0<|x|<1, where AGM is the arithmetic geometric mean.

a(n) = numerator(A054474(n)/(4*16^n)) for n>0.

Prog

(PARI) a(n) = numerator(if(n<0, 0, polcoeff(1-agm(1-x+O(x^(n*2+1)), 1+x+O(x^(n*2+1))), n*2)))
(PARI) a(n) = if(n<1, 0, numerator(Vec(1-1/sum(k=0, 2+2*n, (x^k*binomial(k-(1/2), k))^2)+O(x^(4+2*n)))[1+2*(n-1)]))

Crossrefs

Cf. A056982 (denominators).

Cf. A038534, A038533 (numerator, denominator EllipticK/Pi).

Cf. A054474, A060629.

Sequence in context: A283354 A379702 A156330 * A056253 A101832 A374091

Adjacent sequences: A389634 A389635 A389636 * A389638 A389639 A389640

Keyword

nonn,frac

Author

Thomas Scheuerle, Oct 09 2025

Status

approved