A208681 -id:A208681 - cp's OEIS frontend

A002439 Glaisher's T numbers.

1, 23, 1681, 257543, 67637281, 27138236663, 15442193173681, 11828536957233383, 11735529528739490881, 14639678925928297567703, 22427641105413135505628881, 41393949926819051111431239623, 90592214447886493688036507587681, 231969423543894989257690172433129143

Offset: 0

N. J. A. Sloane

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
From Peter Bala, Dec 18 2021: (Start)
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).
We make the following conjectures:
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).
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.
If true, then for each i the expansion of exp( Sum_{n >= 1} a_i(n)*x^n/n ) has integer coefficients.
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.
  (ii)  a(m*n) == a(m)^n (mod 3^k) for k = 2*v_3(m) + 2.
4)(i)   a(2*m*n) == a(n)^(2*m) (mod 2^k) for k = v_2(m) + 7
  (ii)  a((2*m+1)*n) == a(n)^(2*m+1) (mod 2^k) for k = v_2(m) + 7.
5)(i)   a(3*m*n) == a(n)^(3*m) (mod 3^k) for k = v_3(m) + 2
  (ii)  a((3*m+1)*n) == a(n)^(3*m+1) (mod 3^k) for k = v_3(m) + 2
  (iii) a((3*m+2)*n) == a(n)^(3*m+2) (mod 3^2).
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)

			G.f. = 1 + 23*x + 1681*x^2 +257543*x^3 + 67637281*x^4 + 27138236663*x^5 + ...

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.
J. W. L. Glaisher, Messenger of Math., 28 (1898), 36-79, see esp. p. 76.
J. W. L. Glaisher, On the Bernoullian function, Q. J. Pure Appl. Math., 29 (1898), 1-168.
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
G. E. Andrews, J. Jimenez-Urroz and K. Ono, q-series identities and values of certain L-functions, Duke Math J., 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)
J. Bryson, K. Ono, S. Pitman and R. C. Rhoades, Unimodal Sequences and Quantum and Mock Modular Forms, P. 2.
Frank Garvan, Congruences and relations for r-Fishburn numbers, arXiv:1406.5611 [math.NT], 2014.
J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., 31 (1899), 216-235.
K. Hikami, Volume Conjecture and Asymptotic Expansion of q-Series, Experimental Mathematics Vol. 12, Issue 3 (2003).
Michael E. Hoffman, Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences, The Electronic Journal of Combinatorics, Volume 6.1 (1999): Research paper R21, 13 p.
Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.
J. C. P. Miller, Letter to N. J. A. Sloane, Mar 26 1971
Hjalmar Rosengren, Elliptic pfaffians and solvable lattice models, arXiv preprint arXiv:1605.02915 [math-ph], 2016.
A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938 [math.NT], 2011.
Don Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology, vol.40, pp.945-960 (2001).
Index entries for sequences related to Glaisher's numbers

Cf. A000364, A000464, A002105, A079144, A158690.
Bisections: A156175, A156176.
Twice this sequence gives A000191. A208679, A208680, A208681.

m:=32; R:=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

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:

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 *)
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 *)
a[n_] := -(-4)^n 3^(1 + 2 n) EulerE[1 + 2 n, 1/6]  (* Bill Gosper, Oct 12 2015 *)

{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 */

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

Q_{2n+1}(sqrt(3))/sqrt(3), where the polynomials Q_n() are defined in A104035. - N. J. A. Sloane, Nov 06 2009
E.g.f.: sin(2*x)/(2*cos(3*x)) = Sum a(n)*x^(2*n+1)/(2*n+1)!.
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).
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.
From Peter Bala, Mar 24 2009: (Start)
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.
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)
From Peter Bala, Jan 21 2011: (Start)
Let I = sqrt(-1) and w = exp(2*Pi*I/6). Then
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).
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).
Cf. A002437 and A182825. (End)
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
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
From Peter Bala, May 11 2017: (Start)
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.
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.
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 - ...)))).
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)
a(n) == 23^n ( mod (2^7)*(3^2) ). - Peter Bala, Dec 25 2021

More terms from Michael Somos

Offset changed from 1 to 0 by N. J. A. Sloane, Dec 11 1999

A208679 Kashaev's invariant for the (5,2)-torus knot (Solomon's seal knot).

Original entry on oeis.org

1, 71, 14641, 6242711, 4555133281, 5076970085351, 8024733763147921, 17074591123571719991, 47056485265721520250561, 163059403058191163396938631, 693897612604719894794535433201

Offset: 1

Peter Bala, Mar 01 2012

In general, Kashaev’s invariant for the (2*m+1,2)-torus knot has e.g.f. 1/2*sin(2*x)/cos((2*m+1)*x). Case m = 1 is A002439. For other examples see A208680 and A208681.
From Peter Bala, Dec 20 2021: (Start)
We make the following conjectures:
1) Taking the sequence modulo an integer k gives an eventually periodic sequence with period dividing phi(k). For example, the sequence taken modulo 9 begins [1, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, ...] with an apparent pre-period of length 1 and a period [8, 7, 5, 1, 2, 4] of length 6 = phi(9).
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. (End)

Peter Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)
K. Hikami, Volume Conjecture and Asymptotic Expansion of q-Series, Experimental Mathematics Vol. 12, Issue 3 (2003).

Cf. A002439 ((3,2)-torus knot), A208680, A208681, A208730, A208733, A057569.

Cf. A060096, A060097.

A208679 := proc(n) option remember; if n = 1 then 1; else (-4)^(n-1) - add((-25)^k*binomial(2*n-1,2*k)*procname(n-k),k=1..n) ; end if; end proc:
seq(A208679(n),n = 1..20) # Peter Bala, Dec 20 2021
A208679 := 5*10^(2*n-2)*(-1)^n*euler(2*n-1,3/10):
seq(A208679(n),n = 1..11); # Miles Wilson, Aug 05 2024

nmax = 20; Table[(CoefficientList[Series[1/2*Sin[2*x]/Cos[5*x], {x, 0, 2*nmax}], x] * Range[0, 2*nmax - 1]!)[[j]], {j, 2, 2*nmax + 1, 2}] (* Vaclav Kotesovec, Aug 30 2015 *)

my(x='x+O('x^30), v=Vec(serlaplace((1/2)*sin(2*x)/cos(5*x)))); vector(#v\2,n,v[2*n-1]) \\ Joerg Arndt, Aug 08 2024

E.g.f.: (1/2)*sin(2*x)/cos(5*x) = x + 71*x^3/3! + 14641*x^5/5! + ....
Define F(q) := Sum_{m,n >= 0} (q^(-m*n)*product {i = 1.. m+n} (1-q^i)). For the expansion of F(1-q) and F(exp(-t)) see A208733 and A208730 respectively. Kitami gives the conjectural e.g.f. exp(-9*t)*F(exp(-40*t)) = 1 + 71*t + 14641*t^2/2! + ....
a(n) = (-1)^n/(4*n+4)*20^(2*n+1)*Sum_{k = 1..20} X(k)*B(2*n+2,k/20), where B(n,x) is a Bernoulli polynomial and X(n) is a periodic function modulo 20 given by X(n) = 0 except for X(20*n+3) = X(20*n+17) = 1 and X(20*n+7) = X(20*n+13) = -1.
a(n) = 1/2*(-1)^(n+1)*L(-2*n-1,X) in terms of the associated L-series attached to the periodic arithmetical function X.
a(n) ~ (2*n-1)! * 2^(2*n-3/2) * 5^(2*n-1) * sqrt(5-sqrt(5)) / Pi^(2*n). - Vaclav Kotesovec, Aug 30 2015
From Peter Bala, May 11 2017: (Start)
Let X = 40*x. G.f. with offset 0: A(x) = 1 + 71*x + 14641*x^2 + ... = 1/(1 + 9*x - 2*X/(1 - 3*X/(1 + 9*x - 9*X/(1 - 11*X/(1 + 9*x - 21*X/(1 - 24*X/(1 + 9*x - ...))))))), where the sequence [2, 3, 9, 11, ..., n*(5*n - 1)/2, n*(5*n + 1)/2, ...] of unsigned coefficients in the partial numerators of the continued fraction is A057569.
A(x) = 1/(1 + 49*x - 3*X/(1 - 2*X/(1 + 49*x - 11*X/(1 - 9*X/(1 + 49*x - 24*X/(1 - 21*X/(1 + 49*x - 42*X/(1 - 38*X/(1 + 49*x - ...))))))))), where the sequence [3, 2, 11, 9, 24, 21, ...] of unsigned coefficients in the partial numerators of the continued fraction is obtained by swapping pairs of adjacent terms of A057569. Let B(x) = 1/(1 - 9*x)*A(x/(1 - 9*x)), that is, B(x) is the 9_th binomial transform of A(x). Then B(x/40) = 1 + 2*x + 10*x^2 + 104*x^3 + ... is the o.g.f. for A208730. (End)
From Peter Bala, Dec 20 2021: (Start)
a(1) = 1, a(n) = (-4)^(n-1) - Sum_{k = 1..n} (-25)^k*C(2*n-1,2*k)*a(n-k).
a(n) == 71^(n-1) ( mod (2^7)*3*(5^2) ). (End)
a(n) = 5*10^(2*n - 2)*(-1)^n*E(2*n - 1, 3/10), where E(n,x) is the n-th Euler polynomial in x (A060096/A060097). - Miles Wilson, Aug 05 2024

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A002439 Glaisher's T numbers.

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A208679 Kashaev's invariant for the (5,2)-torus knot (Solomon's seal knot).

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A208680 Kashaev invariant for the (7,2)-torus knot.

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A208732 Sequence related to Kashaev's invariant for the (9,2)-torus knot.

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A156652 Numerator of Euler(n, 7/18).

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A208735 Sequence related to Kashaev's invariant for the (9,2)-torus knot.

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