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%I M5138 N2228 #109 Sep 08 2022 08:44:30
%S 1,23,1681,257543,67637281,27138236663,15442193173681,
%T 11828536957233383,11735529528739490881,14639678925928297567703,
%U 22427641105413135505628881,41393949926819051111431239623,90592214447886493688036507587681,231969423543894989257690172433129143
%N Glaisher's T numbers.
%C Kashaev’s invariant for the (3,2)-torus knot. See Hikami 2003. For other Kashaev invariants see A208679, A208680, and A208681. - _Peter Bala_, Mar 01 2012
%C From _Peter Bala_, Dec 18 2021: (Start)
%C Glaisher's T numbers occur in the evaluation of the L-function L(X_12,s) := Sum_{k >= 1} X_12(k)/k^s for positive even values of s, where X_12(n) = A110161(n) is a nonprincipal Dirichlet character mod 12: the result is L(X_12,2*n+2) = a(n)/(6*sqrt(3)*36^n*(2*n+1)!) * Pi^(2*n+2).
%C We make the following conjectures:
%C 1) Taking the sequence modulo an integer k gives an eventually periodic sequence with period dividing phi(k). For example, the sequence taken modulo 50 begins [1, 23, 31, 43, 31, 13, 31, 33, 31, 3, 31, 23, 31, 43, 31, 13, 31, 33, 31, 3, 31, 23, ...] and appears to have a pre-period of length 1 and a period of length 10 = (1/2)*phi(50).
%C 2) Let i >= 0 and 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.
%C If true, then for each i the expansion of exp( Sum_{n >= 1} a_i(n)*x^n/n ) has integer coefficients.
%C 3)(i) a(m*n) == a(m)^n (mod 2^k) for k = 2*v_2(m) + 7, where v_p(i) denotes the p-adic valuation of i.
%C (ii) a(m*n) == a(m)^n (mod 3^k) for k = 2*v_3(m) + 2.
%C 4)(i) a(2*m*n) == a(n)^(2*m) (mod 2^k) for k = v_2(m) + 7
%C (ii) a((2*m+1)*n) == a(n)^(2*m+1) (mod 2^k) for k = v_2(m) + 7.
%C 5)(i) a(3*m*n) == a(n)^(3*m) (mod 3^k) for k = v_3(m) + 2
%C (ii) a((3*m+1)*n) == a(n)^(3*m+1) (mod 3^k) for k = v_3(m) + 2
%C (iii) a((3*m+2)*n) == a(n)^(3*m+2) (mod 3^2).
%C 6) For prime p >= 5, a((p-1)/2*n*m) == a((p-1)/2*n)^m (mod p^k) for k = v_p(m-1) + 1. (End)
%D A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 76.
%D J. W. L. Glaisher, Messenger of Math., 28 (1898), 36-79, see esp. p. 76.
%D J. W. L. Glaisher, On the Bernoullian function, Q. J. Pure Appl. Math., 29 (1898), 1-168.
%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 T. D. Noe, <a href="/A002439/b002439.txt">Table of n, a(n) for n=0..100</a>
%H G. E. Andrews, J. Jimenez-Urroz and K. Ono, <a href="http://www.math.wisc.edu/~ono/reprints/055.pdf">q-series identities and values of certain L-functions</a>, Duke Math J., 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 J. Bryson, K. Ono, S. Pitman and R. C. Rhoades, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/140.pdf">Unimodal Sequences and Quantum and Mock Modular Forms</a>, P. 2.
%H Frank Garvan, <a href="http://arxiv.org/abs/1406.5611">Congruences and relations for r-Fishburn numbers</a>, arXiv:1406.5611 [math.NT], 2014.
%H J. W. L. Glaisher, <a href="http://plms.oxfordjournals.org/content/s1-31/1/216.extract">On a set of coefficients analogous to the Eulerian numbers</a>, Proc. London Math. Soc., 31 (1899), 216-235.
%H K. Hikami, <a href="http://www.emis.de/journals/EM/expmath/volumes/12/12.3/Hikami.pdf">Volume Conjecture and Asymptotic Expansion of q-Series</a>, Experimental Mathematics Vol. 12, Issue 3 (2003).
%H Michael E. Hoffman, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v6i1r21">Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences</a>, The Electronic Journal of Combinatorics, Volume 6.1 (1999): Research paper R21, 13 p.
%H Hsien-Kuei Hwang and Emma Yu Jin, <a href="https://arxiv.org/abs/1911.06690">Asymptotics and statistics on Fishburn matrices and their generalizations</a>, arXiv:1911.06690 [math.CO], 2019.
%H J. C. P. Miller, <a href="/A002439/a002439_1.pdf">Letter to N. J. A. Sloane, Mar 26 1971</a>
%H Hjalmar Rosengren, <a href="https://arxiv.org/abs/1605.02915">Elliptic pfaffians and solvable lattice models</a>, arXiv preprint arXiv:1605.02915 [math-ph], 2016.
%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.
%H Don Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/">Vassiliev invariants and a strange identity related to the Dedekind eta-function</a>, Topology, vol.40, pp.945-960 (2001).
%H <a href="/index/Ge#Glaisher">Index entries for sequences related to Glaisher's numbers</a>
%F Q_{2n+1)(sqrt(3))/sqrt(3), where the polynomials Q_n() are defined in A104035. - _N. J. A. Sloane_, Nov 06 2009
%F E.g.f.: sin(2*x)/(2*cos(3*x)) = Sum a(n)*x^(2*n+1)/(2*n+1)!.
%F With offset 1 instead of 0: a(1)=1, a(n)=(-4)^(n-1) - Sum_{k=1..n} (-9)^k*C(2*n-1, 2*k)*a(n-k).
%F a(n) = -(-4)^n*3^(2n+1)*E_{2n+1}(1/6), where E is an Euler polynomial. - _Bill Gosper_, Aug 08 2001, corrected Oct 12 2015.
%F From _Peter Bala_, Mar 24 2009: (Start)
%F Basic hypergeometric generating function: exp(-t)*Sum {n = 0..inf} Product {k = 1..n} (1-exp(-24*k*t)) = 1 + 23*t + 1681*t^2/2! + .... For other sequences with generating functions of a similar type see A000364, A000464, A002105, A079144, A158690.
%F a(n) = (1/2)*(-1)^(n+1)*L(-2*n-1), where L(s) is a Dirichlet L-function for a Dirichlet character with modulus 12: L(s) = 1 - 1/5^s - 1/7^s + 1/11^s + - - + .... See the Andrew's link. (End)
%F From _Peter Bala_, Jan 21 2011: (Start)
%F Let I = sqrt(-1) and w = exp(2*Pi*I/6). Then
%F a(n) = I/sqrt(3) *sum {k = 0..2*n+2} w^(n-k) *sum {j = 1..2*n+2} (-1)^(k-j) *binomial(2*n+2,k-j) *(2*j-1)^(2*n+1).
%F This formula can be used to obtain congruences for a(n). For example, for odd prime p we find a(p-1) = 1 (mod p) and a((p-1)/2) = (-1)^((p-1)/2) (mod p).
%F Cf. A002437 and A182825. (End)
%F a(n) = (-1)^n/(4*n+4)*12^(2*n+1)*sum {k = 1..12} X(k)*B(2*n+2,k/12), where B(n,x) is a Bernoulli polynomial and X(n) is a periodic function modulo 12 given by X(n) = 0 except for X(12*n+1) = X(12*n+11) = 1 and X(12*n+5) = X(12*n+7) = -1. - _Peter Bala_, Mar 01 2012
%F a(n) ~ n^(2*n+3/2) * 2^(4*n+3) * 3^(2*n+3/2) / (exp(2*n) * Pi^(2*n+3/2)). - _Vaclav Kotesovec_, Mar 01 2014
%F From _Peter Bala_, May 11 2017: (Start)
%F Let X = 24*x. G.f. A(x) = 1/(1 + x - X/(1 - 2*X/(1 + x - 5*X/(1 - 7*X/(1 + x - 12*X/(1 - ...)))))) = 1 + 23*x + 1681*x^2 + ..., where the sequence [1, 2, 5, 7, 12, ...] of unsigned coefficients in the partial numerators of the continued fraction are generalized pentagonal numbers A001318.
%F A(x) = 1/(1 + 25*x - 2*X/(1 - X/(1 + 25*x - 7*X/(1 - 5*X/(1 + 25*x - 15*X/(1 - 12*X/(1 + 25*x - 26*X/(1 - 22*X/(1 + 25*x - ...))))))))), where the sequence [2, 1, 7, 5, 15, 12, 26, 22, ...] of unsigned coefficients in the partial numerators is obtained by swapping pairs of adjacent generalized pentagonal numbers.
%F G.f. as a J-fraction: A(x) = 1/(1 - 23*x - 2*X^2/(1 - 167*x - 5*7*X^2/(1 - 455*x - 12*15*X^2/(1 - 887*x - ...)))).
%F Let B(x) = 1/(1 - x)*A(x/(1 - x)), that is, B(x) is the binomial transform of A(x). Then B(x/24) is the o.g.f. for A079144. (End)
%F a(n) == 23^n ( mod (2^7)*(3^2) ). - _Peter Bala_, Dec 25 2021
%e G.f. = 1 + 23*x + 1681*x^2 +257543*x^3 + 67637281*x^4 + 27138236663*x^5 + ...
%p A002439 := proc(n) option remember; if n = 0 then 1; else (-4)^n-add((-9)^k*binomial(2*n+1, 2*k)*procname(n-k), k=1..n+1) ; end if; end proc:
%t a[n_] := a[n] = (-4)^n - Sum[(-9)^k*Binomial[2n + 1, 2k]*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 11}] (* _Jean-François Alcover_, Dec 05 2011, after Maple *)
%t With[{nn=30},Take[CoefficientList[Series[Sin[2x]/(2Cos[3x]),{x,0,nn}], x]Range[0,nn-1]!,{2,-1,2}]] (* _Harvey P. Dale_, Feb 05 2012 *)
%t a[n_] := -(-4)^n 3^(1 + 2 n) EulerE[1 + 2 n, 1/6] (* _Bill Gosper_, Oct 12 2015 *)
%o (PARI) {a(n) = my(m=n+1); if( m<2, m>0, (-4)^(m-1) - sum(k=1, m, (-9)^k * binomial(2*m-1, 2*k) * a(n-k)))}; /* _Michael Somos_, Dec 11, 1999 */
%o (Magma) m:=32; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Sin(2*x)/(2*Cos(3*x)) )); [Factorial(2*n-1)*b[2*n-1]: n in [1..Floor((m-2)/2)]]; // _G. C. Greubel_, Jul 04 2019
%o (Sage) m = 32; T = taylor(sin(2*x)/(2*cos(3*x)), x, 0, m); [factorial(2*n+1)*T.coefficient(x, 2*n+1) for n in (0..(m-2)/2)] # _G. C. Greubel_, Jul 04 2019
%Y Cf. A000364, A000464, A002105, A079144, A158690.
%Y Bisections: A156175, A156176.
%Y Twice this sequence gives A000191. A208679, A208680, A208681.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Michael Somos_
%E Offset changed from 1 to 0 by _N. J. A. Sloane_, Dec 11 1999
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