A000464 - OEIS

%I M4812 N2059 #106 Nov 01 2024 11:13:23

%S 1,11,361,24611,2873041,512343611,129570724921,44110959165011,

%T 19450718635716001,10784052561125704811,7342627959965776406281,

%U 6023130568334172003579011,5858598896811701995459355761,6667317162352419006959182803611,8776621742176931117228228227924441

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

%C From _Peter Bala_, Dec 22 2021: (Start)

%C Conjectures:

%C 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.

%C 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.

%C 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.

%C 4)(i)   a(2*m*n) == a(n)^(2*m) (mod 2^k) for k = v_2(m) + 4

%C   (ii)  a((2*m+1)*n) == a(n)^(2*m+1) (mod 2^k) for k = v_2(m) + 4. (End)

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

%D 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.

%D I. J. Schwatt, Intro. to Operations with Series, Chelsea, p. 278.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Matthew House, <a href="/A000464/b000464.txt">Table of n, a(n) for n = 0..215</a> (terms 0..50 from T. D. Noe)

%H G. E. Andrews, J. Jimenez-Urroz, and K. Ono, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/055.pdf">q-series identities and values of certain L-functions</a>, Duke Math Jour., Volume 108, No.3 (2001), 395-419.

%H Peter Bala, <a href="/A002439/a002439.pdf">Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)</a>

%H D. Dumont, <a href="http://dx.doi.org/10.1006/aama.1995.1014">Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers</a>, Adv. Appl. Math., 16 (1995), 275-296.

%H Simon Plouffe, <a href="/A000464/a000464.txt">Table of n, a(n) for n=0..1023</a>

%H D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0223295-5">Generalized Euler and class numbers</a>. Math. Comp. 21 (1967) 689-694.

%H D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1968-0227093-9">Corrigenda to: "Generalized Euler and class numbers"</a>, Math. Comp. 22 (1968), 699

%H D. Shanks, <a href="/A000003/a000003.pdf">Generalized Euler and class numbers</a>, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

%H Alan D. Sokal, <a href="https://arxiv.org/abs/1804.04498">The Euler and Springer numbers as moment sequences</a>, arXiv:1804.04498 [math.CO], 2018.

%H A. Vieru, <a href="http://arxiv.org/abs/1107.2938">Agoh's conjecture: its proof, its generalizations, its analogues</a>, arXiv preprint arXiv:1107.2938 [math.NT], 2011.

%F E.g.f.: Sum_{k>=0} a(k)x^(2k+1)/(2k+1)! = sin(x)/cos(2x).

%F 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]

%F From _Peter Bala_, Mar 24 2009: (Start)

%F Basic hypergeometric generating function:

%F 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.

%F 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)

%F From _Peter Bala_, Jun 18 2009: (Start)

%F 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).

%F 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].

%F 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)

%F 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

%F 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

%F a(n) ~ (2*n+1)! * 2^(4*n+7/2) / Pi^(2*n+2). - _Vaclav Kotesovec_, May 03 2014

%F 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

%F From _Peter Bala_, May 11 2017: (Start)

%F 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) - ...))))))).

%F 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 - ...))))))).

%F 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)

%F 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

%F From _Peter Bala_, Dec 04 2021: (Start)

%F 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.

%F 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)

%F From _Peter Bala_, Dec 22 2021: (Start)

%F a(1) = 1, a(n) = (-1)^(n-1) - Sum_{k = 1..n} (-4)^k*C(2*n-1,2*k)*a(n-k).

%F a(n) == 1 (mod 10); a(5*n+1) == 0 mod(11);

%F a(n) == - 23^(n+1) (mod 108); a(n) == (7^2)*59^n (mod 144);

%F a(n) == 11^n (mod 240); a(n) == (11^2)*131^n (mod 360). (End)

%p a := n -> (-1)^n*2^(6*n+4)*(Zeta(0, -2*n-1, 5/8)-Zeta(0, -2*n-1, 7/8)):

%p seq(a(n), n=0..12); # _Peter Luschny_, Oct 15 2015

%t 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 *)

%t 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 *)

%o (PARI) 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 */

%Y Row 2 of A235606.

%Y Cf. A064073. Bisection of A000822, A001586.

%Y Cf. A000364, A000828, A002105, A002439, A079144, A158690.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_

%E Better description, new reference, Aug 15 1995