A000320 Generalized tangent numbers d(5,n).

4, 272, 55744, 23750912, 17328937984, 19313964388352, 30527905292468224, 64955605537174126592, 179013508069217017790464, 620314831396713435870789632, 2639743384489464189324523208704, 13533573366345611477262311433961472, 82274260343572247169162187576069586944

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Lars Blomberg, Table of n, a(n) for n = 1..189
• D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
• D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699
• D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Crossrefs

Cf. A000318, A000187, A349265, A349264.

Programs

Maple

egf := sec(5*x)*(sin(x) + sin(3*x)): ser := series(egf, x, 26):
seq((2*n-1)!*coeff(ser, x, 2*n-1), n = 1..13); # Peter Luschny, Nov 21 2021

Mathematica

nmax = 15; km0 = 10; Clear[dd]; L[a_, s_, km_] := Sum[JacobiSymbol[-a, 2 k + 1]/(2k+1)^s, {k, 0, km}]; d[a_ /; a>1, n_, km_] := (2n-1)! L[-a, 2n, km] (2a/Pi)^(2n)/Sqrt[a] // Round; dd[km_] := dd[km] = Table[d[5, n, km], {n, 1, nmax}]; dd[km0]; dd[km = 2km0]; While[dd[km] != dd[km/2, km = 2 km]]; A000320 = dd[km] (* Jean-François Alcover, Feb 07 2016 *)

Formula

a(n) = (2*n-1)!*[x^(2*n-1)](sec(5*x)*(sin(x) + sin(3*x))). - Peter Luschny, Nov 21 2021

Extensions

Formula producing A000326, rather than this sequence, deleted by Sean A. Irvine, Sep 09 2010

a(10)-a(13) from Lars Blomberg, Sep 07 2015