A000187 Generalized Euler numbers, c(5,n). (Formerly M2153 N0858)

2, 30, 3522, 1066590, 604935042, 551609685150, 737740947722562, 1360427147514751710, 3308161927353377294082, 10256718523496425979562270, 39490468691102039103925777602, 184856411587530526077816051412830, 1033888847501229495999134528615701122

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]

Formula

From the Shanks paper: Consider the Dirichlet series L_a(s) = sum_{k>=0} (-a|2k+1) / (2k+1)^s, where (-a|2k+1) is the Jacobi symbol. Then the numbers c_(a,n) are defined by L_a(2n+1)= (Pi/(2a))^(2n+1)*sqrt(a)* c(a,n)/ (2n)! for a > 1 and n = 0,1,2,... - Sean A. Irvine, Mar 26 2012

From Peter Bala, Nov 18 2020: (Start)

a(n) = (-1)^n*10^(2*n)*( E(2*n,1/10) + E(2*n,3/10) ), where E(n,x) are the Euler polynomials - see A060096.

Row 5 of A235605.

G.f.: A(x) = 2*cos(x)*cos(3*x)/( 2*cos(x)*cos(4*x) - cos(3*x) ) = 2 + 30*x^2/2! + 3522*x^4/4! + ....

Alternative forms:

A(x) = (exp(i*x) + exp(3*i*x) + exp(7*i*x) + exp(9*i*x))/(1 + exp(10*i*x));

A(x) = (sqrt(5)/10)*( sec(x + Pi/5) + sec(x + 2*Pi/5) - sec(x + 3*Pi/5) - sec(x + 4*Pi/5) ). (End)

a(n) = (2*n)!*[x^(2*n)](sec(5*x)*(cos(2*x) + cos(4*x))). - Peter Luschny, Nov 21 2021

a(n) ~ 2^(4*n + 2) * 5^(2*n + 1/2) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Apr 15 2022

Example

a(3) = 1066590: L_5(7) = Sum_{n >= 0} (-1)^n*( 1/(10*n+1)^7 + 1/(10*n+3)^7 + 1/(10*n+7)^7 + 1/(10*n+9)^7 ) = 1066590*( (1/6!)*sqrt(5)*(Pi/10)^7 ). - Peter Bala, Nov 18 2020

Maple

seq((-1)^n*(10)^(2*n)*(euler(2*n, 1/10) + euler(2*n, 3/10)), n = 0..11); # Peter Bala, Nov 18 2020
egf := sec(5*x)*(cos(2*x) + cos(4*x)): ser := series(egf, x, 26):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..11); # Peter Luschny, Nov 21 2021

Mathematica

a0=5; nmax=20; km0 = nmax; Clear[cc]; L[a_, s_, km_] := Sum[JacobiSymbol[ -a, 2k+1]/(2k+1)^s, {k, 0, km}]; c[a_, n_, km_] := 2^(2n+1)*Pi^(-2n-1)*(2n)!*a^(2n+1/2)*L[a, 2n+1, km] // Round; cc[km_] := cc[km] = Table[ c[a0, n, km], {n, 0, nmax}]; cc[km0]; cc[km = 2km0]; While[cc[km] != cc[ km/2, km = 2km]]; A000187 = cc[km] (* Jean-François Alcover, Feb 05 2016 *)

Crossrefs

Row 5 of A235605.

Cf. A000192, A000490, A060096, A000320, A349265, A349264.

Sequence in context: A220719 A359665 A030249 * A053851 A369681 A077521

Adjacent sequences: A000184 A000185 A000186 * A000188 A000189 A000190

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 02 2000

Status

approved