A392689 a(n) = numerator( (-1)^(n-1)*H(2*n)/(2*n + 1) ), where H(n) is the n-th harmonic number.

0, 1, -5, 7, -761, 671, -6617, 1171733, -143327, 751279, -55835135, 830139, -269564591, 34395742267, -10876020307, 300151059037, -586061125622639, 54062195834749, -1481133119857, 2053580969474233, -50687277590093, 286263092775233, -5884182435213075787

Offset

0,3

References

Konrad Knopp, Theory and application of infinite series, Blackie & Son Limited, London and Glasgow, 1954. See p. 216.

Links

Robert Israel, Table of n, a(n) for n = 0..1147

Index entries for sequences related to harmonic numbers.

Formula

a(n) = numerator( [x^(2*n+1)] arctan(x)*log(1 + x^2)/2 ).

Sum_{n>=1} a(n)/A392690(n) = Pi*log(2)/8 (A102886). - Amiram Eldar, Jul 09 2026

Example

arctan(x)*log(1 + x^2)/2 = (1/2)*x + (-5/12)*x^3 + (7/20)*x^5 + (-761/2520)*x^7 + ...

Maple

H:= 0: R:= 0:
for n from 1 to 30 do
  H:= H + 1/(2*n-1) + 1/(2*n);
  R:= R, numer((-1)^(n-1)*H/(2*n+1));
od:
R; # Robert Israel, Feb 27 2026

Mathematica

a[n_]:=Numerator[(-1)^(n-1)*HarmonicNumber[2n]/(2n+1)]; Array[a, 23, 0]

Prog

(PARI) a(n) = numerator((-1)^(n-1)*sum(k=1, 2*n, 1/k)/(2*n + 1)); \\ Michel Marcus, Jan 21 2026

Crossrefs

Cf. A001008, A005408, A102886, A392690 (denominators).

Sequence in context: A020467 A089344 A114363 * A083687 A101829 A056252

Adjacent sequences: A392686 A392687 A392688 * A392690 A392691 A392692

Keyword

sign,frac,changed

Author

Stefano Spezia, Jan 20 2026

Status

approved