A001587 Generalized Euler numbers.

2, 6, 46, 522, 7970, 152166, 3487246, 93241002, 2849229890, 97949265606, 3741386059246, 157201459863882, 7205584123783010, 357802951084619046, 19133892392367261646, 1096291279711115037162, 67000387673723462963330, 4350684698032741048452486, 299131045427247559446422446

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

These numbers are related to the values at negative integers of the L-functions for two primitive Dirichlet characters of conductor 24. - F. Chapoton, Oct 05 2020

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 = 0..199
• LMFDB, character 24.5
• LMFDB, character 24.11
• D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967) 689-694.
• D. Shanks, Corrigendum: 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

Bisections are A000192 and A000411. Overview in A349264.

Similar sequences: A000111, A225147.

Programs

Maple

egf := sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x)): ser := series(egf, x, 20): seq(n!*coeff(ser, x, n), n = 0..17); # Peter Luschny, Nov 21 2021

Sage

t = PowerSeriesRing(QQ, 't').gen()
f = 2 * (sin(3 * t) + cos(3 * t)) / (2 * cos(4 * t) - 1)
f.egf_to_ogf().list() # F. Chapoton, Oct 06 2020

Formula

E.g.f.: 2 (sin(3 x) + cos(3 x)) / (2 cos(4 x) - 1). - F. Chapoton, Oct 06 2020

a(n) ~ 2^(2*n + 2) * 3^(n + 1/2) * n^(n + 1/2) / (exp(n) * Pi^(n + 1/2)). - Vaclav Kotesovec, Nov 05 2021

a(n) = n!*[x^n](sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x))). - Peter Luschny, Nov 21 2021

Extensions

a(11)-a(14) from Lars Blomberg, Sep 10 2015