A057866 Sum_{k>=1} 1/(tanh(k*Pi) * k^(4n-1)) = Pi^(4n-1)*a(n)/A057867(n).

7, 19, 1453, 13687, 7708537, 4472029801, 149780635937, 11231299844779783, 3688053840923281541, 2659842854283579394387, 1228751826452728351300837, 67537532722660373286810600661, 251492292317888012003479295207, 25465609788816025420512226447159951

Offset

1,1

References

E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford Univ. Press, 1939, p. 135. See Example 15.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..157

Simon Plouffe, Identities inspired by Ramanujan Notebooks (part 2), April 2006.

Linas Vepštas, On Plouffe's Ramanujan identities, The Ramanujan Journal, Vol. 27 (2012), pp. 387-408; arXiv preprint, arXiv:math/0609775 [math.NT], 2006-2010.

Eric Weisstein's World of Mathematics, Riemann Zeta Function.

Formula

Numerator of coefficient of Pi^(4n-1) in Ramanujan-like series for zeta(4n-1).

From Jwalin Bhatt, Apr 15 2026: (Start)

a(n)/A057867(n) = -(2^(4n-2)/(4n)!) * Sum_{j=0..2n} (-1)^j * binomial(4n, 2j) * B_{2j} * B_{4n-2j}, where B_k are the Bernoulli numbers (A027641/A027642).

Limit_{n->oo} (Pi^(4n-1)*a(n)/A057867(n)-coth(Pi)) / (zeta(4n-1)-1) = coth(2*Pi). (End)

a(n) = A394976(2n-1). - Jwalin Bhatt, Apr 28 2026

Example

Sum_{k>=1} 1/(tanh(k*Pi)k^3) = Pi^3*7/180,
Sum_{k>=1} 1/(tanh(k*Pi)k^7) = Pi^7*19/56700.

Mathematica

Numerator[Table[2^(k-1)/(k+1)! Sum[(-1)^(n-1)Binomial[k+1, 2n]BernoulliB[k+1-2n]BernoulliB[2n], {n, 0, (k+1)/2}], {k, 3, 50, 4}]]

Prog

(Python)
from sympy import binomial, bernoulli, factorial
def a(n):
    s = sum(
        (-1)**j * binomial(4*n, 2*j) * bernoulli(4*n-2*j) * bernoulli(2*j)
        for j in range(2*n + 1)
    )
    frac = -2**(4*n-2) * s / factorial(4*n)
    return frac.numerator  # Jwalin Bhatt, Apr 15 2026

Crossrefs

Cf. A027641, A027642, A057867 (denominators), A384962, A394976.

Sequence in context: A330875 A330852 A267237 * A329001 A334982 A339698

Adjacent sequences: A057863 A057864 A057865 * A057867 A057868 A057869

Keyword

nonn,frac

Author

Eric W. Weisstein

Extensions

Definition revised by N. J. A. Sloane, Sep 20 2009, following a suggestion of Michael Somos, Feb 11 2004

Name changed by Jwalin Bhatt, Apr 15 2026

Status

approved