A392685 a(n) = numerator( (1/(4*n + 2))*Sum_{i=0..2*n} (-1)^i/(2*i+1) ).

1, 13, 263, 36979, 622637, 11757173, 1346255081, 116752370597, 3610501179557, 133330680156299, 5457995496252709, 234389556075339277, 77030060483083029083, 4078795356946236329749, 4075520848199724411379, 14657533947750493292463071, 1126808597274538496041637, 75454987789826394512538029

Offset

0,2

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..577

Formula

a(n) = numerator( [x^(4*n+2)] arctan(x)*log((1+x)/(1-x))/4 ).

a(n) = numerator((Pi + 2*LerchPhi(-1, 1, 3/2 + 2*n))/(4*(4*n + 2))).

Example

arctan(x)*log((1+x)/(1-x))/4 = (1/2)*x^2 + (13/90)*x^6 + (263/3150)*x^10 + ...

Maple

R:= 1:
S:= 1:
for n from 1 to 30 do
  S:= S - 1/(4*n-1) + 1/(4*n+1);
  R:= R, numer(1/(4*n+2) * S);
od:
R; # Robert Israel, Feb 27 2026

Mathematica

a[n_]:=Numerator[Sum[(-1)^i/(2i+1), {i, 0, 2n}]/(4n+2)]; Array[a, 18, 0]

Crossrefs

Cf. A016825, A392686 (denominators).

Sequence in context: A142811 A034833 A101004 * A340936 A142931 A142262

Adjacent sequences: A392682 A392683 A392684 * A392686 A392687 A392688

Keyword

nonn,frac

Author

Stefano Spezia, Jan 20 2026

Status

approved