A000191 -id:A000191 - 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

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

Original entry on oeis.org

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

A007289 Expansion of e.g.f. (sin(2x) + cos(x)) / cos(3x).

Original entry on oeis.org

1, 2, 8, 46, 352, 3362, 38528, 515086, 7869952, 135274562, 2583554048, 54276473326, 1243925143552, 30884386347362, 825787662368768, 23657073914466766, 722906928498737152, 23471059057478981762, 806875574817679474688, 29279357851856595135406

Offset: 0

N. J. A. Sloane and Simon Plouffe

nonn

Arises in the enumeration of alternating 3-signed permutations.

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..406 (terms 0..200 from Vincenzo Librandi)
R. Ehrenborg and M. A. Readdy, Sheffer posets and r-signed permutations, Preprint, 1994. (Annotated scanned copy)
Richard Ehrenborg and Margaret A. Readdy, Sheffer posets and r-signed permutations, Annales des Sciences Mathématiques du Québec, 19 (1995), 173-196.

Row 3 of A349271.
Cf. A006873, A007286, A225109, A000191 (bisection), A000436 (bisection).
Cf. A000111, A136630, A352638.

A007289 := proc(n) local k,j; add(add((-1)^j*binomial(k,j)*(k-2*j)^n*I^(n-k),j=0..k),k=0..n) end: # Peter Luschny, Jul 31 2011

mx = 17; Range[0, mx]! CoefficientList[ Series[ (Sin[2 x] + Cos[x])/Cos[3 x], {x, 0, mx}], x] (* Robert G. Wilson v, Apr 28 2013 *)

my(x='x+O('x^66)); Vec(serlaplace((sin(2*x) + cos(x)) / cos(3*x))) \\ Joerg Arndt, Apr 28 2013

from mpmath import mp, polylog, im
mp.dps = 32; mp.pretty = True
def aperm3(n): return 2*((1-I)/(1+I))^n*(1+add(binomial(n,j)*polylog(-j,I)*3^j for j in (0..n)))
def A007289(n) : return im(aperm3(n))
[int(A007289(n)) for n in (0..17)] # Peter Luschny, Apr 28 2013

E.g.f.: (sin(2*x) + cos(x)) / cos(3*x).
a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^j*binomial(k,j)*(k-2*j)^n*I^(n-k). - Peter Luschny, Jul 31 2011
a(n) = Im(2*((1-I)/(1+I))^n*(1+sum_{j=0..n}(binomial(n,j)*Li_{-j}(I)*3^j))). - Peter Luschny, Apr 28 2013
a(n) ~ n! * 2^(n+1)*3^(n+1/2)/Pi^(n+1). - Vaclav Kotesovec, Jun 15 2013
a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022
From Seiichi Manyama, Jun 25 2025: (Start)
E.g.f.: 1/(1 - 2 * sin(x)).
a(n) = Sum_{k=0..n} 2^k * k! * i^(n-k) * A136630(n,k), where i is the imaginary unit. (End)

A000436 Generalized Euler numbers c(3,n).

Original entry on oeis.org

1, 8, 352, 38528, 7869952, 2583554048, 1243925143552, 825787662368768, 722906928498737152, 806875574817679474688, 1118389087843083461066752, 1884680130335630169428983808, 3794717805092151129643367268352

Offset: 0

N. J. A. Sloane

nonn

			G.f. = 1 + 8*x + 352*x^2 + 38528*x^3 + 7869952*x^4 + 2583554048*x^5 + ...

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
Michael E. Hoffman, Derivative polynomials, Euler Polynomials, and associated integer sequences, El. J. Combinat. 6 (see Th. 3.3).
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]

Row 3 of A235605.
Bisections: A156177 and A156178.
Cf. A000191, A007289, overview in A349264.
Cf. A210657, A255882.

A000436 := proc(nmax) local a,n,an; a := [1] : n := 1 : while nops(a)< nmax do an := 1-sum(binomial(2*n,2*i)*3^(2*n-2*i)*(-1)^i*op(i+1,a),i=0..n-1) : a := [op(a),an*(-1)^n] ; n := n+1 ; od ; RETURN(a) ; end:
A000436(10) ; # R. J. Mathar, Nov 19 2006
a := n -> 2*(-144)^n*(Zeta(0,-2*n,1/6)-Zeta(0,-2*n,2/3)):
seq(a(n), n=0..12); # Peter Luschny, Mar 11 2015

a[0] = 1; a[n_] := a[n] = (-1)^n*(1 - Sum[(-1)^i*Binomial[2n, 2i]*3^(2n - 2i)*a[i], {i, 0, n-1}]); Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Jan 31 2012, after R. J. Mathar *)
With[{nn=30},Take[CoefficientList[Series[Cos[x]/Cos[3x],{x,0,nn}], x] Range[ 0,nn]!,{1,-1,2}]] (* Harvey P. Dale, May 22 2012 *)

x='x+O('x^66); v=Vec(serlaplace( cos(x) / cos(3*x) ) ); vector(#v\2,n,v[2*n-1]) \\ Joerg Arndt, Apr 27 2013

from mpmath import mp, lerchphi
mp.dps = 32; mp.pretty = True
def A000436(n): return abs(3^(2*n)*2^(2*n+1)*lerchphi(-1,-2*n,1/3))
[A000436(n) for n in (0..12)]  # Peter Luschny, Apr 27 2013

E.g.f.: cos(x) / cos(3*x) (even powers only).
For n>0, a(n) = A002114(n)*2^(2n+1) = (1/3)*A002112(n)*2^(2n+1). - Philippe Deléham, Jan 17 2004
a(n) = Sum_{k=0..n} (-1)^k*9^(n-k)*A086646(n,k). - Philippe Deléham, Oct 27 2006
(-1)^n a(n) = 1 - Sum_{i=0..n-1} (-1)^i*binomial(2n,2i)*3^(2n-2i)*a(i). - R. J. Mathar, Nov 19 2006
a(n) = P_{2n}(sqrt(3))/sqrt(3) (where the polynomials P_n() are defined in A155100). - N. J. A. Sloane, Nov 05 2009
E.g.f.: E(x) = cos(x)/cos(3*x) = 1 + 4*x^2/(G(0)-2*x^2); G(k) = (2*k+1)*(k+1) - 2*x^2 + 2*x^2*(2*k+1)*(k+1)/G(k+1); (continued fraction, Euler's kind, 1-step). - Sergei N. Gladkovskii, Jan 02 2012
G.f.: 1 / (1 - 2*4*x / (1 - 6*6*x / (1 - 8*10*x / (1 - 12*12*x / (1 - 14*16*x / (1 - 18*18*x / ...)))))). - Michael Somos, May 12 2012
a(n) = | 3^(2*n)*2^(2*n+1)*lerchphi(-1,-2*n,1/3) |. - Peter Luschny, Apr 27 2013
a(n) = (-1)^n*6^(2*n)*E(2*n,1/3), where E(n,x) denotes the n-th Euler polynomial. Calculation suggests that the expansion exp( Sum_{n >= 1} a(n)*x^n/n ) = exp( 8*x + 352*x^2/2 + 38528*x^3/3 + ... ) = 1 + 8*x + 208*x^2 + 14336*x^3 + ... has integer coefficients. Cf. A255882. - Peter Bala, Mar 10 2015
a(n) = 2*(-144)^n*(zeta(-2*n,1/6)-zeta(-2*n,2/3)), where zeta(a,z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015
From Vaclav Kotesovec, May 05 2020: (Start)
For n>0, a(n) = (2*n)! * (zeta(2*n+1, 1/6) - zeta(2*n+1, 5/6)) / (sqrt(3)*Pi^(2*n+1)).
For n>0, a(n) = (-1)^(n+1) * 2^(2*n-1) * Bernoulli(2*n) * (zeta(2*n+1, 1/6) - zeta(2*n+1, 5/6)) / (Pi*sqrt(3)*zeta(2*n)). (End)
Conjecture: for each positive integer k, the sequence defined by a(n) (mod k) is eventually periodic with period dividing phi(k). For example, modulo 13 the sequence becomes [1, 8, 1, 9, 12, 10, 0, 8, 1, 9, 12, 10, 0, ...]; after the initial term 1 this appears to be a periodic sequence of period 6, a divisor of phi(13) = 12. - Peter Bala, Dec 11 2021

A002437 a(n) = A000364(n) * (3^(2*n+1) + 1)/4.

Original entry on oeis.org

1, 7, 305, 33367, 6815585, 2237423527, 1077270776465, 715153093789687, 626055764653322945, 698774745485355051847, 968553361387420436695025, 1632180870878422847476890007, 3286322019402928956112227932705, 7791592461957309952817483706344167, 21485762937086358457367440231243675985

Offset: 0

N. J. A. Sloane

The terms are multiples of the Euler numbers (A000364).

			a(4) = A000364(4) * (3^(2*4+1)+1)/4 = 1385 * (3^9+1)/4 = 1385 * 4921 = 6815585.

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. 75.
J. W. L. Glaisher, Messenger of Math., 28 (1898), 36-79, see esp. p. 51.
L. B. W. Jolley, Summation of Series, Dover, 2nd ed. (1961)
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Michael E. Hoffman, Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences, The Electronic Journal of Combinatorics, vol. 6, no. 1, #R21, (1999).

Bisections: A156168, A156169.
Cf. A000191, A001209, A012494, A000364, A000281, A156134, A349429.
Cf. other sequences with a g.f. of the form cos(x)/(1 - k*sin^2(x)): A012494 (k=-1),  A001209 (k=1/2), A000364 (k=1), A000281 (k=2), A156134 (k=3).

Q:=proc(n) option remember; if n=0 then RETURN(1); else RETURN(expand((u^2+1)*diff(Q(n-1),u)+u*Q(n-1))); fi; end;
[seq(subs(u=sqrt(3),Q(2*n)),n=0..25)];

Table[Abs[EulerE[2 n]] (3^(2 n + 1) + 1) / 4, {n, 0, 30}] (* Vincenzo Librandi, Feb 07 2017 *)

A000364(n) * (3^(2*n+1) + 1)/4.
Q_2n(sqrt(3)), where the polynomials Q_n() are defined in A104035. - N. J. A. Sloane, Nov 06 2009
a(n) = (-1)^n*Sum_{k = 0..2*n-1} w^(2*n+k)*Sum_{j = 1..2*n-1} (-1)^(k-j)*binomial(2*n-1,k-j)*(2*j - 1)^(2*n-2), where  w = exp(2*Pi*i/6) (i = sqrt(-1)). Cf. A002439. - Peter Bala, Jan 21 2011
Sum_{n>=1} (-1)^floor((n-1)/2) 1/A007310(n)^s = r_s with r_{2s+1} = 2 *(Pi/6)^(2s+1) *a(s) /(2s)!. [Jolley eq (315)]. - R. J. Mathar, Mar 24 2011
From Peter Bala, Feb 06 2017: (Start)
E.g.f.: cos(x)^2/cos(3*x) = cos(x)/(1 - 4*sin(x)^2) = 1 + 7*x^2/2! + 305*x^4/4! + 33367*x^6/6! + .... This is the even part of (1/2)*sec(x + Pi/3). Cf. A000191. (End)
a(n) = (1/2)*Integral_{x = 0..inf} x^(2*n)*cosh(Pi*x/3)/cosh(Pi*x/2) dx. - Cf. A000281. - Peter Bala, Nov 08 2019

More terms from Herman P. Robinson

Further terms from N. J. A. Sloane, Nov 06 2009

A000192 Generalized Euler numbers c(6,n).

Original entry on oeis.org

2, 46, 7970, 3487246, 2849229890, 3741386059246, 7205584123783010, 19133892392367261646, 67000387673723462963330, 299131045427247559446422446, 1658470810032820740402966226850, 11179247066648898992009055586869646, 90035623994788132387893239340761189570

Offset: 0

N. J. A. Sloane

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).

Sean A. Irvine, Table of n, a(n) for n = 0..250
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]
Eric Weisstein's World of Mathematics, Euler Number.

Row 6 of A235605.

Cf. A000187, A000191, A000411, A001587, A349264.

egf := sec(6*x)*(cos(x) + cos(5*x)): ser := series(egf, x, 24):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..10); # Peter Luschny, Nov 21 2021

L[ a_, s_, t_:10000 ] := Plus@@Table[ N[ JacobiSymbol[ -a, 2k+1 ](2k+1)^(-s), 30 ], {k, 0, t} ]; c[ a_, n_, t_:10000 ] := (2n)!/Sqrt[ a ](2a/Pi)^(2n+1)L[ a, 2n+1, t ] (* Eric W. Weisstein, Aug 30 2001 *)

t = PowerSeriesRing(QQ, 't', default_prec=24).gen()
f = 2 * cos(3 * t) / (2 * cos(4 * t) - 1)
f.egf_to_ogf().list()[::2] # F. Chapoton, Oct 06 2020

E.g.f.: 2*cos(3*x) / (2*cos(4*x) - 1). - F. Chapoton, Oct 06 2020
a(n) = (2*n)!*[x^(2*n)](sec(6*x)*(cos(x) + cos(5*x))). - Peter Luschny, Nov 21 2021
a(n) ~ 2^(6*n + 5/2) * 3^(2*n + 1/2) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Apr 15 2022

More terms from Eric W. Weisstein, Aug 30 2001

A235606 Shanks's array d_{a,n} (a >= 1, n >= 1) that generalizes the tangent numbers, read by antidiagonals upwards.

Original entry on oeis.org

1, 1, 2, 2, 11, 16, 4, 46, 361, 272, 4, 128, 3362, 24611, 7936, 6, 272, 16384, 515086, 2873041, 353792, 8, 522, 55744, 4456448, 135274562, 512343611, 22368256, 8, 904, 152166, 23750912, 2080374784, 54276473326, 129570724921, 1903757312, 12, 1408, 355688

Offset: 1

N. J. A. Sloane, Jan 22 2014

			The array begins:
A000182: 1,  2,    16,      272,        7936,         353792, ...
A000464: 1, 11,   361,    24611,     2873041,      512343611, ...
A000191: 2, 46,  3362,   515086,   135274562,    54276473326, ...
A000318: 4,128, 16384,  4456448,  2080374784,  1483911200768, ...
A000320: 4,272, 55744, 23750912, 17328937984, 19313964388352, ...
A000411: 6,522,152166, 93241002, 97949265606,157201459863882, ...
A064072: 8,904,355688,296327464,423645846728,925434038426824, ...
...

D. Shanks. "Generalized Euler and Class Numbers." Math. Comput. 21, 689-694, 1967. Math. Comput. 22, 699, 1968.

Lars Blomberg, Table of n, a(n) for n = 1..5050
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Rows: A000182 (tangent numbers), A000464, A000191, A000318, A000320, A000411, A064072-A064075, ...
Columns: A000061, A000176, A000488, A000518, ...
Cf. A235605.

amax = nmax = 10; km0 = 10; Clear[dd]; L[a_, s_, km_] := Sum[JacobiSymbol[ -a, 2k+1]/(2k+1)^s, {k, 0, km}]; d[1, n_, km_] := 2(2n-1)! L[-1, 2n, km] (2/Pi)^(2n) // Round; d[a_ /; a>1, n_, km_] := (2n-1)! L[-a, 2n, km] (2a/ Pi)^(2n)/Sqrt[a] // Round; dd[km_] := dd[km] = Table[d[a, n, km], {a, 1, amax}, {n, 1, nmax}]; dd[km0]; dd[km = 2km0]; While[dd[km] != dd[km/2, km = 2km]]; A235606 = dd[km]; Table[A235606[[ a-n+1, n]], {a, 1, amax}, {n, 1, a}] // Flatten (* Jean-François Alcover, Feb 05 2016 *)
dds[b_, nm_] := With[{ns = Range[nm]}, (-1)^(ns - 1) If[Mod[b, 4] == 1, Sum[JacobiSymbol[k, b] (b - 4 k)^(2 ns - 1), {k, 1, (b - 1)/2}], Sum[JacobiSymbol[b, 2 k + 1] (b - (2 k + 1))^(2 ns - 1), {k, 0, (b - 2)/2}]]];
dsfs[1, nm_] := dsfs[1, nm] = (2 Range[nm] - 1)! CoefficientList[Series[Tan[x], {x, 0, 2 nm - 1}]/x, x^2];
dsfs[b_, nm_] := dsfs[b, nm] = Fold[Function[{ds, dd}, Append[ds, dd - Sum[ds[[-i]] (-b^2)^i Binomial[2 Length[ds] + 1, 2 i], {i, Length[ds]}]]], {}, dds[b, nm]];
rowA235606[a_, nm_] := With[{facs = FactorInteger[a], ns = Range[nm]}, With[{b = Times @@ (#^Mod[#2, 2] &) @@@ facs}, If[a == b, dsfs[b, nm], If[b == 1, 1/2, 1] dsfs[b, nm] Sqrt[a/b]^(4 ns - 1) Times @@ Cases[facs, {p_, e_} /; p > 2 && e > 1 :> 1 - JacobiSymbol[b, p]/p^(2 ns)]]]];
arr = Table[rowA235606[a, 10], {a, 10}];
Flatten[Table[arr[[r - n + 1, n]], {r, Length[arr]}, {n, r}]] (* Matthew House, Oct 30 2024 *)

Shanks gives recurrences.

More terms from Lars Blomberg, Sep 07 2015

A000318 Generalized tangent numbers d(4,n).

Original entry on oeis.org

4, 128, 16384, 4456448, 2080374784, 1483911200768, 1501108249821184, 2044143848640217088, 3605459138582973251584, 7995891855149741436305408, 21776918737280678860353961984, 71454103701490016776039304265728, 278008871543597996197497752082448384

Offset: 1

N. J. A. Sloane

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 = 1..100
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]
Eric Weisstein's World of Mathematics, Padé approximants.

Cf. A000182, A000191, A000490, A349264.

egf := sec(4*x)*sin(4*x): ser := series(egf, x, 26):
seq((2*n-1)!*coeff(ser, x, 2*n-1), n = 1..12); # Peter Luschny, Nov 21 2021

nn = 30; t = Rest@Union[Range[0, nn - 1]! CoefficientList[Series[Tan[x], {x, 0, nn}], x]]; t2 = t*2^Range[2, 2*nn, 4] (* T. D. Noe, Jun 19 2012 *)

a(n) = 2^(4n-2) * A000182(n).
The g.f. has the following continued fraction expansion: g.f. = [4, b(0), c(0), b(1), c(1), b(2), c(2), ...] where b(n) = (Sum_{k=0..n} 1/(2*k+1))^2 / (128*(n+1)*x), c(n) = -4/((2*n+3)*(Sum_{k=0..n} 1/(2*k+1))*(Sum_{k=0..n+1} 1/(2*k+1))) and each convergent of this continued fraction is a Padé approximant of the Maclaurin series Sum_{k>=1} a(n)*x^(n-1). - Thomas Baruchel, Oct 19 2005
a(n) = (2*n-1)!*[x^(2*n-1)](sec(4*x)*sin(4*x)). - Peter Luschny, Nov 21 2021

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 03 2000

A225147 a(n) = Im((1-I)^(1-n)*A_{n, 3}(I)) where A_{n, k}(x) are the generalized Eulerian polynomials.

Original entry on oeis.org

-1, 2, 5, -46, -205, 3362, 22265, -515086, -4544185, 135274562, 1491632525, -54276473326, -718181418565, 30884386347362, 476768795646785, -23657073914466766, -417370516232719345, 23471059057478981762, 465849831125196593045, -29279357851856595135406

Offset: 0

Peter Luschny, Apr 30 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Peter Luschny, Generalized Eulerian polynomials.

Cf. A000810 (real part (up to sign)), A212435 (k=2), A122045 (k=1), A002439.

Bisections are A002438 and A000191.

B := proc(n, u, k) option remember;
if n = 1 then if (u < 0) or (u >= 1) then 0 else 1 fi
else k*u*B(n-1, u, k) + k*(n-u)*B(n-1, u-1, k) fi end:
EulerianPolynomial := proc(n, k, x) local m; if x = 0 then RETURN(1) fi;
add(B(n+1, m+1/k, k)*u^m, m = 0..n); subs(u=x, %) end:
seq(Im((1-I)^(1-n)*EulerianPolynomial(n, 3, I)), n=0..19);

CoefficientList[Series[-E^(-2*x)*Sech[3*x],{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Sep 29 2014 after Sergei N. Gladkovskii *)
Table[-6^n EulerE[n,1/6], {n,0,19}] (* Peter Luschny, Nov 16 2016 after Peter Bala *)

from mpmath import mp, polylog, im
mp.dps = 32; mp.pretty = True
def A225147(n): return im(-2*I*(1+add(binomial(n,j)*polylog(-j,I)*3^j for j in (0..n))))
[int(A225147(n)) for n in (0..19)]

a(n) = Im(-2*i*(1+Sum_{j=0..n}(binomial(n,j)*Li{-j}(i)*3^j))).
For a recurrence see the Maple program.
G.f.: conjecture -T(0)/(1+2*x), where T(k) = 1 - 9*x^2*(k+1)^2/(9*x^2*(k+1)^2 + (1+2*x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2013
a(n) = -(-3)^n*skp(n, 2/3), where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 19 2014
G.f.: A225147 = -1/T(0), where T(k) = 1 + 2*x + (k+1)^2*(3*x)^2/ T(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 29 2014
E.g.f.: -exp(-2*x)*sech(3*x). - Sergei N. Gladkovskii, Sep 29 2014
a(n) ~ n! * (sqrt(3)*sin(Pi*n/2) - cos(Pi*n/2)) * 2^(n+1) * 3^n / Pi^(n+1). - Vaclav Kotesovec, Sep 29 2014
From Peter Bala, Nov 13 2016: (Start)
a(n) = - 6^n*E(n,1/6), where E(n,x) denotes the Euler polynomial of order n.
a(2*n) = (-1)^(n+1)*A002438(n); a(2*n+1) = (1/2)*(-1)^n*A002439(n). (End)

A262144 Square array read by antidiagonals upwards: the n-th row o.g.f. is exp( Sum_{i >= 1} d(n,i+1)*x^i/i ) for n >= 1, where d(n,k) is Shanks's array of generalized Euler and class numbers.

Original entry on oeis.org

1, 1, 2, 1, 11, 10, 1, 46, 241, 108, 1, 128, 2739, 10411, 2214, 1, 272, 16384, 265244, 836321, 75708, 1, 522, 64964, 2883584, 45094565, 112567243, 3895236, 1, 904, 212325, 18852096, 822083584, 12975204810, 22949214033

Offset: 1

Peter Bala, Sep 18 2015

Shanks's array d(n,k) n >= 1, k >= 1, is A235606.
We conjecture that the entries of the present array are all integers. More generally, we conjecture that for r = 1, 2, ... and for each n >= 1, the expansion of exp( Sum_{i >= 1} d(n,i + r)*x^i/i ) has integer coefficients. This is the case r = 1.
For the similarly defined array associated with Shanks' c(n,k) array see A262143.

			The triangular array begins
1
1   2
1  11     10
1  46    241      108
1 128   2739    10411      2214
1 272  16384   265244    836321       75708
1 522  64964  2883584  45094565   112567243     3895236
1 904 212325 18852096 822083584 12975204810 22949214033 ...
The square array begins (row indexing n starts at 1)
1, 2, 10, 108, 2214, 75708, 3895236, 280356120, 26824493574, ...
1, 11, 241, 10411, 836321, 112567243, 22949214033, 6571897714923, 2507281057330113, ...
1, 46, 2739, 265244, 45094565, 12975204810, 5772785327575, 3656385436507960, 3107332328608143945, ...
1, 128, 16384, 2883584, 822083584, 395136991232, 300338473074688, 330739694704787456, 493338658405976375296, ...
1, 272, 64864, 18852096, 8133183744, 5766226378752, 6562478680375296, 11019751545852395520, 25333348417380699340800, ...
1, 522, 212325, 94501768, 57064909374, 54459242196516, 84430282319806062, 197625548666434041000, 642556291067409622713543, ...
1, 904, 586452, 382674008, 311514279098, 379982635729752, 753288329161251844, 2308779464340711480136, 10003494921382094286802995, ...

P. Bala, Notes on logarithmic differentiation, the binomial transform and series reversion
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, April 2012.
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]

Cf. A000182 (d(1,n)), A000464 (d(2,n)), A000191 (d(3,n)), A000318 (d(4,n)), A000320 (d(5,n)), A000411 (d(6,n)), A064072 (d(7,n)), A235605, A235606, A262143, A262145 (row 1 of square array).

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A002439 Glaisher's T numbers.

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

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A007289 Expansion of e.g.f. (sin(2*x) + cos(x)) / cos(3*x).

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A000436 Generalized Euler numbers c(3,n).

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A002437 a(n) = A000364(n) * (3^(2*n+1) + 1)/4.

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A000192 Generalized Euler numbers c(6,n).

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

A007289 Expansion of e.g.f. (sin(2x) + cos(x)) / cos(3x).