A208679 -id:A208679 - 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

A208681 Kashaev's invariant for the (9,2)-torus knot.

Original entry on oeis.org

1, 239, 160801, 222359759, 525750911041, 1898604115708079, 9723130520022672481, 67030256200148854573199, 598528825179130480174293121, 6719801498668147110144664875119, 92651189588518508157161032926540961

Offset: 1

Peter Bala, Mar 01 2012

Compare with A156652. For other Kashaev invariants see A002439, A208679 and A208680.
From Peter Bala, Dec 25 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 21 begins [1, 8, 4, 20, 13, 17, 1, 8, 4, 20, 13, 17, 1, 8, 4, 20, 13, 17, ...], which appears to be a purely periodic sequence with period 6 = (1/2)*phi(21).
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), A156652, A208679, A208680.

A208681 := proc(n) option remember; if n = 1 then 1; else (-4)^(n-1) - add((-81)^k*binomial(2*n-1,2*k)*procname(n-k),k=1..n) ; end if; end proc:
seq(A208681(n),n = 1..20) # Peter Bala, Dec 25 2021

a[n_] := (2n-1)! SeriesCoefficient[(1/2)(Sin[2x]/ Cos[9x]), {x, 0, 2n-1}];
Table[a[n], {n, 1, 11}] (* Jean-François Alcover, Sep 23 2022 *)

E.g.f.: 1/2*sin(2*x)/cos(9*x) = x + 239*x^3/3! + 160801*x^5/5! + ....
a(n) = (-1)^n/(4*n+4)*36^(2*n+1)*sum {k = 1..36} X(k)*B(2*n+2,k/36), where B(n,x) is a Bernoulli polynomial and X(n) is a periodic function modulo 36 given by X(n) = 0 except for X(36*n+7) = X(36*n+29) = 1 and X(36*n+11) = X(36*n+25) = -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.
From Peter Bala, May 16 2017: (Start)
O.g.f. (with offset 0) as continued fraction: A(x) = 1/(1 + 49*x - 8*36*x/(1 - 10*36*x/(1 + 49*x -...- n*(9*n-1)*36*x/(1 - n*(9*n+1)*36*x/(1 + 49*x - ... ))))).
Also,  A(x) = 1/(1 + 121*x - 10*36*x/(1 - 8*36*x/(1 + 121*x -...- n*(9*n+1)*36*x/(1 - n*(9*n-1)*36*x/(1 + 121*x - ... ))))). (End)
a(n) ~ sin(Pi/9) * 2^(4*n) * 3^(4*n-2) * n^(2*n-1/2) / (Pi^(2*n-1/2) * exp(2*n)). - Vaclav Kotesovec, May 18 2017
From Peter Bala, Dec 25 2021: (Start)
a(1) = 1, a(n) = (-4)^(n-1) - Sum_{k = 1..n} (-81)^k*C(2*n-1,2*k)*a(n-k).
a(n) == 239^(n-1) mod ( (2^8)*(3^4)*5 ). (End)

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

Original entry on oeis.org

1, 143, 58081, 48571823, 69471000001, 151763444497103, 470164385248041121, 1960764928973430783983, 10591336845363318048877441, 71933835058256664782546056463, 599982842750416411984319126244961

Offset: 1

Peter Bala, Mar 01 2012

Compare with A156370. For other Kashaev invariants see A002439 ((3,2)-torus knot), A208679 and A208681.

P. 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), A156370, A208679, A208681.

E.g.f.: 1/2*sin(2*x)/cos(7*x) = x + 143*x^3/3! + 58081*x^5/5! + ....
a(n) = (-1)^n/(4*n+4)*28^(2*n+1)*sum {k = 1..28} X(k)*B(2*n+2,k/28), where B(n,x) is a Bernoulli polynomial and X(n) is a periodic function modulo 28 given by X(n) = 0 except for X(28*n+5) = X(28*n+23) = 1 and X(28*n+9) = X(28*n+19) = -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.
From Peter Bala, May 16 2017: (Start)
O.g.f. as continued fraction: A(x) = 1/(1 + 25*x - 6*28*x/(1 - 8*28*x/(1 + 25*x -...- n*(7*n-1)*28*x/(1 - n*(7*n+1)*28*x/(1 + 25*x - ... ))))).
Also,  A(x) = 1/(1 + 81*x - 8*28*x/(1 - 6*28*x/(1 + 81*x -...- n*(7*n+1)*28*x/(1 - n*(7*n-1)*28*x/(1 + 81*x - ... ))))). (End)
a(n) ~ sin(Pi/7) * 2^(4*n) * 7^(2*n-1) * n^(2*n-1/2) / (Pi^(2*n-1/2) * exp(2*n)). - Vaclav Kotesovec, May 18 2017

A208730 Sequence related to Kashaev's invariant for the (5,2)-torus knot.

Original entry on oeis.org

1, 2, 10, 104, 1870, 51632, 2027470, 107354144, 7370645950, 636754087472, 67591284235630, 8647294709864384, 1312197219579059230, 233025643830843282512

Offset: 0

Peter Bala, Mar 01 2012

This is sequence b_n(5) in Table 2 of Hikami 2003.

K. Hikami, Volume Conjecture and Asymptotic Expansion of q-Series, Experimental Mathematics Vol. 12, Issue 3 (2003).

Cf. A079144, A208679, A208731, A208732, A208733.

Define F(q) := sum {m,n >= 0} (q^(-m*n)*product {i = 1.. m+n} (1-q^i)).
E.g.f.: F(exp(-t)) = 1 + 2*t + 10*t^2! + 104*t^3/3! + .... For the expansion of F(1-q) see A208733. F(q) also appears in a conjectural e.g.f. for A208679.
a(n) = (9/40)^n*sum {k = 0..n} binomial(n,k)*A208679(k+1)/9^k.
Conjectural S-fraction for the o.g.f.: 1/(1-2*x/(1-3*x/(1-9*x/(1-11*x/(1-...-1/2*n*(5*n-1)*x/(1-1/2*n*(5*n+1)*x/(1- ....

A208733 Sequence related to Kashaev's invariant for the (5,2)-torus knot.

Original entry on oeis.org

1, 2, 6, 23, 109, 621, 4149, 31851, 276408, 2676388, 28608866, 334647768, 4252140022, 58322409086, 858871301034

Offset: 0

Peter Bala, Mar 02 2012

This is sequence a_n(5) in Table 3 of Hikami 2003.

K. Hikami, Volume Conjecture and Asymptotic Expansion of q-Series, Experimental Mathematics Vol. 12, Issue 3 (2003).

Cf. A208679, A208730, A208734, A209735.

Define F(q) := sum {m,n >= 0} (q^(-m*n)*product {i = 1.. m+n} (1-q^i)).O.g.f.: F(1-q) = 1 + 2*q + 6*q^2 + 23*q^3 + ....

For n >=1, a(n) = 1/n!*sum {k = 1..n} |Stirling1(n,k)|*A208730(k).

A156285 Numerator of Euler(n, 3/10).

Original entry on oeis.org

1, -1, -21, 71, 2541, -14641, -771561, 6242711, 437721081, -4555133281, -399147115101, 5076970085351, 533834157267621, -8024733763147921, -984414289344742641, 17074591123571719991, 2393808447230495554161, -47056485265721520250561

Offset: 0

N. J. A. Sloane, Nov 07 2009

The (unsigned) odd-indexed terms of the sequence appear to give A208679. - Peter Bala, May 16 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200

For denominators see A156276. Cf. A208679.

Numerator[EulerE[Range[0,20],3/10]] (* Vincenzo Librandi, May 05 2012 *)

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A002439 Glaisher's T numbers.

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A208681 Kashaev's invariant for the (9,2)-torus knot.

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

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A208730 Sequence related to Kashaev's invariant for the (5,2)-torus knot.

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A208733 Sequence related to Kashaev's invariant for the (5,2)-torus knot.

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A156285 Numerator of Euler(n, 3/10).

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