A235605 -id:A235605 - cp's OEIS frontend

A000281 Expansion of cos(x)/cos(2x).

1, 3, 57, 2763, 250737, 36581523, 7828053417, 2309644635483, 898621108880097, 445777636063460643, 274613643571568682777, 205676334188681975553003, 184053312545818735778213457, 193944394596325636374396208563

Offset: 0

N. J. A. Sloane

a(n) is (2n)! times the coefficient of x^(2n) in the Taylor series for cos(x)/cos(2x).

			cos x / cos 2*x = 1 + 3*x^2/2 + 19*x^4/8 + 307*x^6/80 + ...

J. W. L. Glaisher, "On the coefficients in the expansions of cos x / cos 2x and sin x / cos 2x", Quart. J. Pure and Applied Math., 45 (1914), 187-222.
I. J. Schwatt, Intro. to Operations with Series, Chelsea, p. 278.
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).

G. C. Greubel, Table of n, a(n) for n = 0..215 (terms 0..50 from T. D. Noe)
P. Bala, A triangle for calculating A000281
P. Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)
Kwang-Wu Chen, An Interesting Lemma for Regular C-fractions, J. Integer Seqs., Vol. 6, 2003.
D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.
Matthieu Josuat-Vergès and Jang Soo Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, arXiv:1101.5608 [math.CO] (2011).
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694, sequence c(2,n).
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]

Cf. A000364 A086646, A002437.
Cf. A064069. Bisection of A000825, A001586.
Cf. A098432. Cf. A093954, A188458, A212435, A235605, A255883.

a := n -> (-1)^n*2^(6*n+1)*(Zeta(0,-2*n,1/8)-Zeta(0,-2*n,5/8)):
seq(a(n), n=0..13); # Peter Luschny, Mar 11 2015

With[{nn=30},Take[CoefficientList[Series[Cos[x]/Cos[2x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Oct 06 2011 *)

{a(n) = if( n<0, 0, n*=2; n! * polcoeff( cos(x + x * O(x^n)) / cos(2*x + x * O(x^n)), n))}; /* Michael Somos, Feb 09 2006 */

a(n) = Sum_{k=0..n} (-1)^k*binomial(2n, 2k)*A000364(n-k)*4^(n-k). - Philippe Deléham, Jan 26 2004
E.g.f.: Sum_{k>=0} a(k)x^(2k)/(2k)! = cos(x)/cos(2x).
a(n-1) is approximately 2^(4*n-3)*(2*n-1)!*sqrt(2)/((Pi^(2*n-1))*(2*n-1)). The approximation is quite good a(250) is of the order of 10^1181 and this formula is accurate to 238 digits. - Simon Plouffe, Jan 31 2007
G.f.: 1 / (1 - 1*3*x / (1 - 4*4*x / (1 - 5*7*x / (1 - 8*8*x / (1 - 9*11*x / ... ))))). - Michael Somos, May 12 2012
G.f.: 1/E(0) where E(k) = 1 - 3*x - 16*x*k*(2*k+1) - 16*x^2*(k+1)^2*(4*k+1)*(4*k+3)/E(k+1) (continued fraction, 1-step). - Sergei N. Gladkovskii, Sep 17 2012
G.f.: T(0)/(1-3*x), where T(k) = 1 - 16*x^2*(4*k+1)*(4*k+3)*(k+1)^2/( 16*x^2*(4*k+1)*(4*k+3)*(k+1)^2 - (32*x*k^2+16*x*k+3*x-1 )*(32*x*k^2+80*x*k+51*x -1)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 11 2013
From Peter Bala, Mar 09 2015: (Start)
a(n) = (-1)^n*4^(2*n)*E(2*n,1/4), where E(n,x) denotes the n-th Euler polynomial.
O.g.f.: Sum_{n >= 0} 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 + x*(4*k + 1)^2) = 1 + 3*x + 57*x^2 + 2763*x^3 + ....
We appear to have the asymptotic expansion Pi/(2*sqrt(2)) - Sum {k = 0..n - 1} (-1)^floor(k/2)/(2*k + 1) ~ 1/(2*n) - 3/(2*n)^3 + 57/(2*n)^5 - 2763/(2*n)^7 + .... See A093954.
Bisection of A001586. See also A188458 and A212435. Second row of A235605 (read as a square array).
The expansion of exp( Sum_{n >= 1} a(n)*x^n/n ) appears to have integer coefficients. See A255883. (End)
From Peter Luschny, Mar 11 2015: (Start)
a(n) = ((-64)^n/((n+1/2)))*(B(2*n+1,7/8)-B(2*n+1,3/8)), B(n,x) Bernoulli polynomials.
a(n) = 2*(-16)^n*LerchPhi(-1, -2*n, 1/4).
a(n) = (-1)^n*Sum_{0..2*n} 2^k*C(2*n,k)*E(k), E(n) the Euler secant numbers A122045.
a(n) = (-4)^n*SKP(2*n,1/2) where SKP are the Swiss-Knife polynomials A153641.
a(n) = (-1)^n*2^(6*n+1)*(Zeta(-2*n,1/8) - Zeta(-2*n,5/8)), where Zeta(a,z) is the generalized Riemann zeta function. (End)
From Peter Bala, May 13 2017: (Start)
G.f.: 1/(1 + x - 4*x/(1 - 12*x/(1 + x - 40*x/(1 - 56*x/(1 + x - ... - 4*n(4*n - 3)*x/(1 - 4*n(4*n - 1)*x/(1 + x - ...
G.f.: 1/(1 + 9*x - 12*x/(1 - 4*x/(1 + 9*x - 56*x/(1 - 40*x/(1 + 9*x - ... - 4*n(4*n - 1)*x/(1 - 4*n(4*n - 3)*x/(1 + 9*x - .... (End)
From Peter Bala, Nov 08 2019: (Start)
a(n) = sqrt(2)*4^n*Integral_{x = 0..inf} x^(2*n)*cosh(Pi*x/2)/cosh(Pi*x) dx. Cf. A002437.
The L-series 1 + 1/3^(2*n+1) - 1/5^(2*n+1) - 1/7^(2*n+1) + + - - ... = sqrt(2)*(Pi/4)^(2*n+1)*a(n)/(2*n)! (see Shanks), which gives a(n) ~ (1/sqrt(2))*(2*n)!*(4/Pi)^(2*n+1). (End)

A000436 Generalized Euler numbers c(3,n).

Original entry on oeis.org

1, 8, 352, 38528, 7869952, 2583554048, 1243925143552, 825787662368768, 722906928498737152, 806875574817679474688, 1118389087843083461066752, 1884680130335630169428983808, 3794717805092151129643367268352

Offset: 0

N. J. A. Sloane

nonn

			G.f. = 1 + 8*x + 352*x^2 + 38528*x^3 + 7869952*x^4 + 2583554048*x^5 + ...

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).

T. D. Noe, Table of n, a(n) for n = 0..100
Michael E. Hoffman, Derivative polynomials, Euler Polynomials, and associated integer sequences, El. J. Combinat. 6 (see Th. 3.3).
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]

Row 3 of A235605.
Bisections: A156177 and A156178.
Cf. A000191, A007289, overview in A349264.
Cf. A210657, A255882.

A000436 := proc(nmax) local a,n,an; a := [1] : n := 1 : while nops(a)< nmax do an := 1-sum(binomial(2*n,2*i)*3^(2*n-2*i)*(-1)^i*op(i+1,a),i=0..n-1) : a := [op(a),an*(-1)^n] ; n := n+1 ; od ; RETURN(a) ; end:
A000436(10) ; # R. J. Mathar, Nov 19 2006
a := n -> 2*(-144)^n*(Zeta(0,-2*n,1/6)-Zeta(0,-2*n,2/3)):
seq(a(n), n=0..12); # Peter Luschny, Mar 11 2015

a[0] = 1; a[n_] := a[n] = (-1)^n*(1 - Sum[(-1)^i*Binomial[2n, 2i]*3^(2n - 2i)*a[i], {i, 0, n-1}]); Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Jan 31 2012, after R. J. Mathar *)
With[{nn=30},Take[CoefficientList[Series[Cos[x]/Cos[3x],{x,0,nn}], x] Range[ 0,nn]!,{1,-1,2}]] (* Harvey P. Dale, May 22 2012 *)

x='x+O('x^66); v=Vec(serlaplace( cos(x) / cos(3*x) ) ); vector(#v\2,n,v[2*n-1]) \\ Joerg Arndt, Apr 27 2013

from mpmath import mp, lerchphi
mp.dps = 32; mp.pretty = True
def A000436(n): return abs(3^(2*n)*2^(2*n+1)*lerchphi(-1,-2*n,1/3))
[A000436(n) for n in (0..12)]  # Peter Luschny, Apr 27 2013

E.g.f.: cos(x) / cos(3*x) (even powers only).
For n>0, a(n) = A002114(n)*2^(2n+1) = (1/3)*A002112(n)*2^(2n+1). - Philippe Deléham, Jan 17 2004
a(n) = Sum_{k=0..n} (-1)^k*9^(n-k)*A086646(n,k). - Philippe Deléham, Oct 27 2006
(-1)^n a(n) = 1 - Sum_{i=0..n-1} (-1)^i*binomial(2n,2i)*3^(2n-2i)*a(i). - R. J. Mathar, Nov 19 2006
a(n) = P_{2n}(sqrt(3))/sqrt(3) (where the polynomials P_n() are defined in A155100). - N. J. A. Sloane, Nov 05 2009
E.g.f.: E(x) = cos(x)/cos(3*x) = 1 + 4*x^2/(G(0)-2*x^2); G(k) = (2*k+1)*(k+1) - 2*x^2 + 2*x^2*(2*k+1)*(k+1)/G(k+1); (continued fraction, Euler's kind, 1-step). - Sergei N. Gladkovskii, Jan 02 2012
G.f.: 1 / (1 - 2*4*x / (1 - 6*6*x / (1 - 8*10*x / (1 - 12*12*x / (1 - 14*16*x / (1 - 18*18*x / ...)))))). - Michael Somos, May 12 2012
a(n) = | 3^(2*n)*2^(2*n+1)*lerchphi(-1,-2*n,1/3) |. - Peter Luschny, Apr 27 2013
a(n) = (-1)^n*6^(2*n)*E(2*n,1/3), where E(n,x) denotes the n-th Euler polynomial. Calculation suggests that the expansion exp( Sum_{n >= 1} a(n)*x^n/n ) = exp( 8*x + 352*x^2/2 + 38528*x^3/3 + ... ) = 1 + 8*x + 208*x^2 + 14336*x^3 + ... has integer coefficients. Cf. A255882. - Peter Bala, Mar 10 2015
a(n) = 2*(-144)^n*(zeta(-2*n,1/6)-zeta(-2*n,2/3)), where zeta(a,z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015
From Vaclav Kotesovec, May 05 2020: (Start)
For n>0, a(n) = (2*n)! * (zeta(2*n+1, 1/6) - zeta(2*n+1, 5/6)) / (sqrt(3)*Pi^(2*n+1)).
For n>0, a(n) = (-1)^(n+1) * 2^(2*n-1) * Bernoulli(2*n) * (zeta(2*n+1, 1/6) - zeta(2*n+1, 5/6)) / (Pi*sqrt(3)*zeta(2*n)). (End)
Conjecture: for each positive integer k, the sequence defined by a(n) (mod k) is eventually periodic with period dividing phi(k). For example, modulo 13 the sequence becomes [1, 8, 1, 9, 12, 10, 0, 8, 1, 9, 12, 10, 0, ...]; after the initial term 1 this appears to be a periodic sequence of period 6, a divisor of phi(13) = 12. - Peter Bala, Dec 11 2021

A000192 Generalized Euler numbers c(6,n).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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).

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.

Row 6 of A235605.

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

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

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 *)

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

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

More terms from Eric W. Weisstein, Aug 30 2001

A000187 Generalized Euler numbers, c(5,n).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			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

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).

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]

Row 5 of A235605.

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

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

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 *)

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

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

A235606 Shanks's array d_{a,n} (a >= 1, n >= 1) that generalizes the tangent numbers, read by antidiagonals upwards.

Original entry on oeis.org

1, 1, 2, 2, 11, 16, 4, 46, 361, 272, 4, 128, 3362, 24611, 7936, 6, 272, 16384, 515086, 2873041, 353792, 8, 522, 55744, 4456448, 135274562, 512343611, 22368256, 8, 904, 152166, 23750912, 2080374784, 54276473326, 129570724921, 1903757312, 12, 1408, 355688

Offset: 1

N. J. A. Sloane, Jan 22 2014

			The array begins:
A000182: 1,  2,    16,      272,        7936,         353792, ...
A000464: 1, 11,   361,    24611,     2873041,      512343611, ...
A000191: 2, 46,  3362,   515086,   135274562,    54276473326, ...
A000318: 4,128, 16384,  4456448,  2080374784,  1483911200768, ...
A000320: 4,272, 55744, 23750912, 17328937984, 19313964388352, ...
A000411: 6,522,152166, 93241002, 97949265606,157201459863882, ...
A064072: 8,904,355688,296327464,423645846728,925434038426824, ...
...

D. Shanks. "Generalized Euler and Class Numbers." Math. Comput. 21, 689-694, 1967. Math. Comput. 22, 699, 1968.

Lars Blomberg, Table of n, a(n) for n = 1..5050
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Rows: A000182 (tangent numbers), A000464, A000191, A000318, A000320, A000411, A064072-A064075, ...
Columns: A000061, A000176, A000488, A000518, ...
Cf. A235605.

amax = nmax = 10; km0 = 10; Clear[dd]; L[a_, s_, km_] := Sum[JacobiSymbol[ -a, 2k+1]/(2k+1)^s, {k, 0, km}]; d[1, n_, km_] := 2(2n-1)! L[-1, 2n, km] (2/Pi)^(2n) // Round; d[a_ /; a>1, n_, km_] := (2n-1)! L[-a, 2n, km] (2a/ Pi)^(2n)/Sqrt[a] // Round; dd[km_] := dd[km] = Table[d[a, n, km], {a, 1, amax}, {n, 1, nmax}]; dd[km0]; dd[km = 2km0]; While[dd[km] != dd[km/2, km = 2km]]; A235606 = dd[km]; Table[A235606[[ a-n+1, n]], {a, 1, amax}, {n, 1, a}] // Flatten (* Jean-François Alcover, Feb 05 2016 *)
dds[b_, nm_] := With[{ns = Range[nm]}, (-1)^(ns - 1) If[Mod[b, 4] == 1, Sum[JacobiSymbol[k, b] (b - 4 k)^(2 ns - 1), {k, 1, (b - 1)/2}], Sum[JacobiSymbol[b, 2 k + 1] (b - (2 k + 1))^(2 ns - 1), {k, 0, (b - 2)/2}]]];
dsfs[1, nm_] := dsfs[1, nm] = (2 Range[nm] - 1)! CoefficientList[Series[Tan[x], {x, 0, 2 nm - 1}]/x, x^2];
dsfs[b_, nm_] := dsfs[b, nm] = Fold[Function[{ds, dd}, Append[ds, dd - Sum[ds[[-i]] (-b^2)^i Binomial[2 Length[ds] + 1, 2 i], {i, Length[ds]}]]], {}, dds[b, nm]];
rowA235606[a_, nm_] := With[{facs = FactorInteger[a], ns = Range[nm]}, With[{b = Times @@ (#^Mod[#2, 2] &) @@@ facs}, If[a == b, dsfs[b, nm], If[b == 1, 1/2, 1] dsfs[b, nm] Sqrt[a/b]^(4 ns - 1) Times @@ Cases[facs, {p_, e_} /; p > 2 && e > 1 :> 1 - JacobiSymbol[b, p]/p^(2 ns)]]]];
arr = Table[rowA235606[a, 10], {a, 10}];
Flatten[Table[arr[[r - n + 1, n]], {r, Length[arr]}, {n, r}]] (* Matthew House, Oct 30 2024 *)

Shanks gives recurrences.

More terms from Lars Blomberg, Sep 07 2015

A349271 Array A(n, k) that generalizes Euler numbers, class numbers, and tangent numbers, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 4, 8, 11, 5, 2, 4, 16, 46, 57, 16, 1, 6, 30, 128, 352, 361, 61, 2, 8, 46, 272, 1280, 3362, 2763, 272, 2, 8, 64, 522, 3522, 16384, 38528, 24611, 1385, 2, 12, 96, 904, 7970, 55744, 249856, 515086, 250737, 7936

Offset: 1

Peter Luschny, Nov 23 2021

			Seen as an array:
[1] 1,  1,   1,    2,     5,      16,       61,        272, ... [A000111]
[2] 1,  1,   3,   11,    57,     361,     2763,      24611, ... [A001586]
[3] 1,  2,   8,   46,   352,    3362,    38528,     515086, ... [A007289]
[4] 1,  4,  16,  128,  1280,   16384,   249856,    4456448, ... [A349264]
[5] 2,  4,  30,  272,  3522,   55744,  1066590,   23750912, ... [A349265]
[6] 2,  6,  46,  522,  7970,  152166,  3487246,   93241002, ... [A001587]
[7] 1,  8,  64,  904, 15872,  355688,  9493504,  296327464, ... [A349266]
[8] 2,  8,  96, 1408, 29184,  739328, 22634496,  806453248, ... [A349267]
[9] 2, 12, 126, 2160, 49410, 1415232, 48649086, 1951153920, ... [A349268]
.
Seen as a triangle:
[1] 1;
[2] 1, 1;
[3] 1, 1,  1;
[4] 1, 2,  3,   2;
[5] 2, 4,  8,  11,    5;
[6] 2, 4, 16,  46,   57,    16;
[7] 1, 6, 30, 128,  352,   361,    61;
[8] 2, 8, 46, 272, 1280,  3362,  2763,   272;
[9] 2, 8, 64, 522, 3522, 16384, 38528, 24611, 1385;

William Y. C. Chen, Neil J. Y. Fan, and Jeffrey Y. T. Jia, The generating function for the Dirichlet series Lm(s), Mathematics of Computation, Vol. 81, No. 278, pp. 1005-1023, April 2012.
Ruth Lawrence and Don Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J. Math. 3 (1999), no. 1, 93-107.
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]

A235605 (array generalized Euler secant numbers).
A235606 (array generalized Euler tangent numbers).
A349264 (overview generating functions).
Rows: A000111, A001586, A007289, A349265, A001587, A349266, A349267, A349268.
Columns: A000003 (class numbers), A000061, A000233, A000176, A000362, A000488, A000508, A000518.
Cf. A349263 (main diagonal).

A000233 Generalized class numbers c_(n,1).

Original entry on oeis.org

1, 3, 8, 16, 30, 46, 64, 96, 126, 158, 216, 256, 302, 396, 448, 512, 636, 702, 792, 960, 1052, 1118, 1344, 1472, 1550, 1866, 1944, 2048, 2442, 2540, 2688, 3072, 3212, 3388, 3888, 4032, 4094, 4746, 4928, 5056, 5832, 5852, 5976, 6912, 7020, 7180, 8064, 8192

Offset: 1

N. J. A. Sloane

Let L_a(s) = Sum_{k>=0} (-a|2k+1) /(2k+1)^s be a Dirichlet series, where (-a|2k+1) is the Jacobi symbol. Then the c_(a,n) are defined by L_a(2n+1) = (Pi/(2a))^(2n+1)*sqrt(a)*c_(a,n)/(2n)! for n=0,1,2,..., a=1,2,3,...

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).

Matthew House, Table of n, a(n) for n = 1..10000
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]

Cf. A000508, A000362.

amax = 50; nmax = 1; km0 = 10; Clear[cc]; L[a_, s_, km_] := Sum[ JacobiSymbol[-a, 2 k + 1]/(2 k + 1)^s, {k, 0, km}]; c[1, n_, km_] := 2 (2 n)! L[1, 2 n + 1, km] (2/Pi)^(2 n + 1) // Round; c[a_ /; a > 1, n_, km_] := (2 n)! L[a, 2 n + 1, km] (2 a/Pi)^(2 n + 1)/Sqrt[a] // Round; cc[km_] := cc[km] = Table[c[a, n, km], {a, 1, amax}, {n, 0, nmax}]; cc[km0]; cc[ km = 2 km0]; While[cc[km] != cc[km/2, km = 2 km]]; A000233 = cc[km][[All, 2]] (* Jean-François Alcover, Feb 06 2016 *)
Table[rowA235605[n, 1][[2]], {n, 50}] (* see A235605 *) (* Matthew House, Oct 05 2024 *)

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

Name clarified by James C. McMahon, Nov 30 2023

A000490 Generalized Euler numbers c(4,n).

Original entry on oeis.org

1, 16, 1280, 249856, 90767360, 52975108096, 45344872202240, 53515555843342336, 83285910482761809920, 165262072909347030040576, 407227428060372417275494400, 1219998300294918683087199010816, 4366953142363907901751614431559680, 18406538229888710811704852978971181056

Offset: 0

N. J. A. Sloane

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).

Matthew House, Table of n, a(n) for n = 0..194
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]

Row 4 of A235605.

Cf. A000187, A000436, A000318, A349264.

egf := sec(4*x): ser := series(egf, x, 26):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..11); # Peter Luschny, Nov 21 2021

a0 = 4; nmax = 20; km0 = nmax; Clear[cc]; L[a_, s_, km_] := Sum[ JacobiSymbol[-a, 2*k+1]/(2*k+1)^s, {k, 0, km}]; c[a_, n_, km_] := 2^(2*n +1)*Pi^(-(2*n)-1)*(2*n)!*a^(2*n+1/2)*L[a, 2*n+1, km] // Round; cc[km_] := cc[km] = Table[c[a0, n, km], {n, 0, nmax}]; cc[km0]; cc[km = 2 km0]; While[cc[km] != cc[km/2, km = 2 km]]; A000490 = cc[km] (* Jean-François Alcover, Feb 05 2016 *)
Range[0, 26, 2]! CoefficientList[Series[Sec[4 x], {x, 0, 26}], x^2] (* Matthew House, Oct 05 2024 *)

a(n) = A000364(n)*16^n. - Philippe Deléham, Oct 27 2006

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

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

A064068 Generalized Euler number c(7,n).

Original entry on oeis.org

1, 64, 15872, 9493504, 10562158592, 18878667833344, 49488442978598912, 178867627497727197184, 852509723495811705208832, 5180564635674867885905281024, 39094622102339738427522497380352, 358686739310560735577543742129700864, 3931974790759726002374736527410407145472

Offset: 0

Eric W. Weisstein, Aug 31 2001

Matthew House, Table of n, a(n) for n = 0..180
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.
Eric Weisstein's World of Mathematics, Euler Number.

Row 7 of A235605.

Cf. A064072, A349266, A349264.

egf := sec(7*x)*(cos(x) + cos(3*x) - cos(5*x)): ser := series(egf, x, 24):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..11); # Peter Luschny, Nov 21 2021

Range[0, 24, 2]! CoefficientList[Series[Sec[7 x] (Cos[x] + Cos[3 x] - Cos[5 x]), {x, 0, 24}], x^2] (* Matthew House, Oct 25 2024 *)

a(n) = (2*n)!*[x^(2*n)](sec(7*x)*(cos(x) + cos(3*x) - cos(5*x))). - Peter Luschny, Nov 21 2021

A064069 Generalized Euler number c(8,n).

Original entry on oeis.org

2, 96, 29184, 22634496, 32864600064, 76717014122496, 262665886073094144, 1239981021847665770496, 7719096548270543600615424, 61267211781784116552580202496, 603881788505747521507846892027904, 7236592671961544936200760521440362496, 103612803724706836868168667250308188995584

Offset: 0

Eric W. Weisstein, Aug 31 2001

Matthew House, Table of n, a(n) for n = 0..176
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.
Eric Weisstein's World of Mathematics, Euler Number.

Row 8 of A235605.

Cf. A000281, A064073, A349267, A349264.

egf := sec(8*x)*2*cos(4*x): ser := series(egf, x, 24):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..11); # Peter Luschny, Nov 21 2021

Range[0, 24, 2]! CoefficientList[Series[2 Sec[8 x] Cos[4 x], {x, 0, 24}], x^2] (* Matthew House, Oct 25 2024 *)

a(n) = 2^(4n+5) * A000281(n).

a(n) = (2*n)!*[x^(2*n)](sec(8*x)*2*cos(4*x)). - Peter Luschny, Nov 21 2021

cp's OEIS Frontend

A000281 Expansion of cos(x)/cos(2x).

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A000436 Generalized Euler numbers c(3,n).

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A000192 Generalized Euler numbers c(6,n).

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A000187 Generalized Euler numbers, c(5,n).

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A235606 Shanks's array d_{a,n} (a >= 1, n >= 1) that generalizes the tangent numbers, read by antidiagonals upwards.

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A349271 Array A(n, k) that generalizes Euler numbers, class numbers, and tangent numbers, read by ascending antidiagonals.

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A000233 Generalized class numbers c_(n,1).

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