A000192 Generalized Euler numbers c(6,n). (Formerly M2167 N0865)

2, 46, 7970, 3487246, 2849229890, 3741386059246, 7205584123783010, 19133892392367261646, 67000387673723462963330, 299131045427247559446422446, 1658470810032820740402966226850, 11179247066648898992009055586869646, 90035623994788132387893239340761189570

Offset

0,1

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

Sean A. Irvine, Table of n, a(n) for n = 0..250

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]

Eric Weisstein's World of Mathematics, Euler Number.

Formula

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

a(n) = (2*n)!*[x^(2*n)](sec(6*x)*(cos(x) + cos(5*x))). - Peter Luschny, Nov 21 2021

a(n) ~ 2^(6*n + 5/2) * 3^(2*n + 1/2) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Apr 15 2022

Maple

egf := sec(6*x)*(cos(x) + cos(5*x)): ser := series(egf, x, 24):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..10); # Peter Luschny, Nov 21 2021

Mathematica

L[ a_, s_, t_:10000 ] := Plus@@Table[ N[ JacobiSymbol[ -a, 2k+1 ](2k+1)^(-s), 30 ], {k, 0, t} ]; c[ a_, n_, t_:10000 ] := (2n)!/Sqrt[ a ](2a/Pi)^(2n+1)L[ a, 2n+1, t ] (* Eric W. Weisstein, Aug 30 2001 *)

Prog

(Sage)
t = PowerSeriesRing(QQ, 't', default_prec=24).gen()
f = 2 * cos(3 * t) / (2 * cos(4 * t) - 1)
f.egf_to_ogf().list()[::2] # F. Chapoton, Oct 06 2020

Crossrefs

Row 6 of A235605.

Cf. A000187, A000191, A000411, A001587, A349264.

Sequence in context: A074041 A277554 A000191 * A196197 A273380 A124690

Adjacent sequences: A000189 A000190 A000191 * A000193 A000194 A000195

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Eric W. Weisstein, Aug 30 2001

Status

approved