A064073 -id:A064073 - cp's OEIS frontend

A349264 Generalized Euler numbers, a(n) = n![x^n](sec(4x)(sin(4x) + 1)).

1, 4, 16, 128, 1280, 16384, 249856, 4456448, 90767360, 2080374784, 52975108096, 1483911200768, 45344872202240, 1501108249821184, 53515555843342336, 2044143848640217088, 83285910482761809920, 3605459138582973251584, 165262072909347030040576, 7995891855149741436305408

Offset: 0

Peter Luschny, Nov 20 2021

nonn

			Exponential generating functions of generalized Euler numbers in context:
egf1 = sec(1*x)*(sin(x) + 1).
   [A000111, A000364, A000182]
egf2 = sec(2*x)*(sin(x) + cos(x)).
   [A001586, A000281, A000464]
egf3 = sec(3*x)*(sin(2*x) + cos(x)).
   [A007289, A000436, A000191]
egf4 = sec(4*x)*(sin(4*x) + 1).
   [A349264, A000490, A000318]
egf5 = sec(5*x)*(sin(x) + sin(3*x) + cos(2*x) + cos(4*x)).
   [A349265, A000187, A000320]
egf6 = sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x)).
   [A001587, A000192, A000411]
egf7 = sec(7*x)*(-sin(2*x) + sin(4*x) + sin(6*x) + cos(x) + cos(3*x) - cos(5*x)).
   [A349266, A064068, A064072]
egf8 = sec(8*x)*2*(sin(4*x) + cos(4*x)).
   [A349267, A064069, A064073]
egf9 = sec(9*x)*(4*sin(3*x) + 2)*cos(3*x)^2.
   [A349268, A064070, A064074]

Matthew House, Table of n, a(n) for n = 0..389
William Y. C. Chen, Neil J. Y. Fan, 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]

Row 4 of A349271.
Cf. A000111, A001586, A007289, A349265, A001587, A349266, A349267, A349268.
Secant  bisections: A000364, A000281, A000436, A000490, A000187, A000192, A064068, A064069, A064070.
Tangent bisections: A000182, A000464, A000191, A000318, A000320, A000411, A064072, A064073, A064074.
Cf. A225147, A349429.

sec(4*x)*(sin(4*x) + 1): series(%, x, 20): seq(n!*coeff(%, x, n), n = 0..19);

m = 19; CoefficientList[Series[Sec[4*x] * (Sin[4*x] + 1), {x, 0, m}], x] * Range[0, m]! (* Amiram Eldar, Nov 20 2021 *)

seq(n)={my(x='x + O('x^(n+1))); Vec(serlaplace((sin(4*x) + 1)/cos(4*x)))} \\ Andrew Howroyd, Nov 20 2021

A000464 Expansion of e.g.f. sin(x)/cos(2*x).

Original entry on oeis.org

1, 11, 361, 24611, 2873041, 512343611, 129570724921, 44110959165011, 19450718635716001, 10784052561125704811, 7342627959965776406281, 6023130568334172003579011, 5858598896811701995459355761, 6667317162352419006959182803611, 8776621742176931117228228227924441

Offset: 0

N. J. A. Sloane

From Peter Bala, Dec 22 2021: (Start)
Conjectures:
1) Taking the sequence (a(n))n>=1 modulo an integer k gives a purely periodic sequence with period dividing phi(k). For example, the sequence taken modulo 21 begins [11, 4, 20, 10, 17, 1, 11, 4, 20, 10, 17, 1, 11, 4, 20, 10, 17, 1, ...] with an apparent period of length 6, which divides phi(21) = 12.
2) For i >= 0, define a_i(n) = a(n+i). Then for each i the Gauss congruences a_i(n*p^k) == a_i(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k. If true, then for each i the expansion of exp(Sum_{n >= 1} a_i(n)*x^n/n) has integer coefficients.
3) a(m*n) == a(m)^n (mod 2^k) for k = 2*v_2(m) + 4, where v_p(i) denotes the p-adic valuation of i.
4)(i)   a(2*m*n) == a(n)^(2*m) (mod 2^k) for k = v_2(m) + 4
  (ii)  a((2*m+1)*n) == a(n)^(2*m+1) (mod 2^k) for k = v_2(m) + 4. (End)

H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag.
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).

Matthew House, Table of n, a(n) for n = 0..215 (terms 0..50 from T. D. Noe)
G. E. Andrews, J. Jimenez-Urroz, and K. Ono, q-series identities and values of certain L-functions, Duke Math Jour., Volume 108, No.3 (2001), 395-419.
Peter Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)
D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.
Simon Plouffe, Table of n, a(n) for n=0..1023
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]
Alan D. Sokal, The Euler and Springer numbers as moment sequences, arXiv:1804.04498 [math.CO], 2018.
A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938 [math.NT], 2011.

Row 2 of A235606.
Cf. A064073. Bisection of A000822, A001586.
Cf. A000364, A000828, A002105, A002439, A079144, A158690.

a := n -> (-1)^n*2^(6*n+4)*(Zeta(0, -2*n-1, 5/8)-Zeta(0, -2*n-1, 7/8)):
seq(a(n), n=0..12); # Peter Luschny, Oct 15 2015

With[{nn=30},Take[CoefficientList[Series[Sin[x]/Cos[2x],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* Harvey P. Dale, Mar 23 2012 *)
nmax = 15; km0 = 10; d[n_, km_] := Round[(2^(4n-1/2) (2n-1)! Sum[ JacobiSymbol[2, 2k+1]/(2k+1)^(2n), {k, 0, km}])/Pi^(2n)]; dd[km_] := dd[km] = Table[d[n, km], {n, 1, nmax}]; dd[km0]; dd[km = 2*km0]; While[dd[km] != dd[km/2, km = 2*km]]; A000464 = dd[km] (* Jean-François Alcover, Feb 08 2016 *)

a(n)=if(n<0, 0, n+=n+1; n!*polcoeff(sin(x+x*O(x^n))/cos(2*x+x*O(x^n)),n)) /* Michael Somos, Feb 09 2006 */

E.g.f.: Sum_{k>=0} a(k)x^(2k+1)/(2k+1)! = sin(x)/cos(2x).
a(n) = (-1)^n*L(X,-2n+1) where L(X,z) is the Dirichlet L-function L(X,z) = Sum_{k>=0} X(k)/k^z and where X(k) is the Dirichlet character Legendre(k,2) which begins 1,0,-1,0,-1,0,1,0,1,0,-1,0,-1,0,1,0,1,0,-1,0.... - Benoit Cloitre, Mar 22 2009 [This Dirichlet character is A091337. - Jianing Song, Oct 22 2023]
From Peter Bala, Mar 24 2009: (Start)
Basic hypergeometric generating function:
2*exp(-t)*Sum_{n = 0..inf} (Product_{k = 1..n} (1-exp(-16*k*t))/Product_{k = 1..n+1} (1+exp(-(16*k-8)*t))) = 1 + 11*t + 361*t^2/2! + 24611*t^3/3! + .... For other sequences with generating functions of a similar type see A000364, A002105, A002439, A079144 and A158690.
a(n) = (-1)^(n+1)*L(-2*n-1), where L(s) is the Dirichlet L-function L(s) = 1 - 1/3^s - 1/5^s + 1/7^s + - - + ... [Andrews et al., Theorem 5]. (End)
From Peter Bala, Jun 18 2009: (Start)
a(n) = (-1)^n*B_(2*n+2)(X)/(2*n+2), where B_n(X) denotes the X-Bernoulli number with X a Dirichlet character modulus 8 given by X(8*n+1) = X(8*n+7) = 1, X(8*n+3) = X(8*n+5) = -1 and X(2*n) = 0. See A161722 for the values of B_n(X).
For the theory and properties of the generalized Bernoulli numbers B_n(X) and the associated generalized Bernoulli polynomials B_n(X,x) see [Cohen, Section 9.4].
The present sequence also occurs in the evaluation of the finite sum of powers Sum_{i = 0..m-1} {(8*i+1)^n - (8*i+3)^n - (8*i+5)^n + (8*i+7)^n}, n = 1,2,... - see A151751 for details. (End)
G.f. 1/G(0) where G(k) = 1 + x - x*(4*k+3)*(4*k+4)/(1 - (4*k+4)*(4*k+5)*x/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 11 2012
G.f.: 1/E(0) where E(k) = 1 - 11*x - 32*x*k*(k+1) - 16*x^2*(k+1)^2*(4*k+3)*(4*k+5)/E(k+1) (continued fraction, 1-step). - Sergei N. Gladkovskii, Sep 17 2012
a(n) ~ (2*n+1)! * 2^(4*n+7/2) / Pi^(2*n+2). - Vaclav Kotesovec, May 03 2014
a(n) = (-1)^n*2^(6*n+4)*(Zeta(-2*n-1,5/8)-Zeta(-2*n-1,7/8)). - Peter Luschny, Oct 15 2015
From Peter Bala, May 11 2017: (Start)
G.f. A(x) = 1 + 11*x + 361*x^2 + ... = 1/(1 + x - 12*x/(1 - 20*x/(1 + x - 56*x/(1 - 72*x/(1 + x - ... - 4*n*(4*n - 1)*x/(1 - 4*n*(4*n + 1)*x/((1 + x) - ...))))))).
A(x) = 1/(1 + 9*x - 20*x/(1 - 12*x/(1 + 9*x - 72*x/(1 - 56*x/(1 + 9*x - ... - 4*n*(4*n + 1)*x/(1 - 4*n*(4*n - 1)*x/(1 + 9*x - ...))))))).
It follows that the first binomial transform of A(x) and the ninth binomial transform of A(x) have continued fractions of Stieltjes-type (S-fractions). (End)
a(n) = (-1)^(n+1)*4^(2*n+1)*E(2*n+1,1/4), where E(n,x) is the n-th Euler polynomial. Cf. A002439. - Peter Bala, Aug 13 2017
From Peter Bala, Dec 04 2021: (Start)
F(x) = exp(x)*(exp(2*x) - 1)/(exp(4*x) + 1) = x - 11*x^3/3! + 361x^5/5! - 24611*x^7/7! + ... is the e.g.f. for the sequence [1, 0, -11, 0, 361, 0, -24611, 0, ...], a signed and aerated version of this sequence.
The binomial transform exp(x)*F(x) = x + 2*x^2/2! - 8*x^3/3! - 40*x^4/4! + + - - is an e.g.f. for a signed version of A000828 (omitting the initial term). (End)
From Peter Bala, Dec 22 2021: (Start)
a(1) = 1, a(n) = (-1)^(n-1) - Sum_{k = 1..n} (-4)^k*C(2*n-1,2*k)*a(n-k).
a(n) == 1 (mod 10); a(5*n+1) == 0 mod(11);
a(n) == - 23^(n+1) (mod 108); a(n) == (7^2)*59^n (mod 144);
a(n) == 11^n (mod 240); a(n) == (11^2)*131^n (mod 360). (End)

Better description, new reference, Aug 15 1995

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A349264 Generalized Euler numbers, a(n) = n!*[x^n](sec(4*x)*(sin(4*x) + 1)).

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A000464 Expansion of e.g.f. sin(x)/cos(2*x).

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A064069 Generalized Euler number c(8,n).

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A349267 Generalized Euler numbers, a(n) = n!*[x^n](sec(8*x)*2*(sin(4*x) + cos(4*x))).

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A349264 Generalized Euler numbers, a(n) = n![x^n](sec(4x)(sin(4x) + 1)).

A349267 Generalized Euler numbers, a(n) = n![x^n](sec(8x)2(sin(4x) + cos(4x))).