A002445 -id:A002445 - cp's OEIS frontend

A185896 Triangle of coefficients of (1/sec^2(x))*D^n(sec^2(x)) in powers of t = tan(x), where D = d/dx.

1, 0, 2, 2, 0, 6, 0, 16, 0, 24, 16, 0, 120, 0, 120, 0, 272, 0, 960, 0, 720, 272, 0, 3696, 0, 8400, 0, 5040, 0, 7936, 0, 48384, 0, 80640, 0, 40320, 7936, 0, 168960, 0, 645120, 0, 846720, 0, 362880, 0, 353792, 0, 3256320, 0, 8951040, 0, 9676800, 0, 3628800

Offset: 0

Peter Bala, Feb 07 2011

DEFINITION
Define polynomials R(n,t) with t = tan(x) by
... (d/dx)^n sec^2(x) = R(n,tan(x))*sec^2(x).
The first few are
... R(0,t) = 1
... R(1,t) = 2*t
... R(2,t) = 2 + 6*t^2
... R(3,t) = 16*t + 24*t^3.
This triangle shows the coefficients of R(n,t) in ascending powers of t called the tangent number triangle in [Hodges and Sukumar].
The polynomials R(n,t) form a companion polynomial sequence to Hoffman's two polynomial sequences - P(n,t) (A155100), the derivative polynomials of the tangent and Q(n,t) (A104035), the derivative polynomials of the secant. See also A008293 and A008294.
COMBINATORIAL INTERPRETATION
A combinatorial interpretation for the polynomial R(n,t) as the generating function for a sign change statistic on certain types of signed permutation can be found in [Verges].
A signed permutation is a sequence (x_1,x_2,...,x_n) of integers such that {|x_1|,|x_2|,...|x_n|} = {1,2...,n}. They form a group, the hyperoctahedral group of order 2^n*n! = A000165(n), isomorphic to the group of symmetries of the n dimensional cube.
Let x_1,...,x_n be a signed permutation.
Then 0,x_1,...,x_n,0 is a snake of type S(n;0,0) when 0 < x_1 > x_2 < ... 0.
For example, 0 4 -3 -1 -2 0 is a snake of type S(4;0,0).
Let sc be the number of sign changes through a snake
... sc = #{i, 1 <= i <= n-1, x_i*x_(i+1) < 0}.
For example, the snake 0 4 -3 -1 -2 0  has sc = 1. The polynomial R(n,t) is the generating function for the sign change statistic on snakes of type S(n+1;0,0):
... R(n,t) = sum {snakes in S(n+1;0,0)} t^sc.
See the example section below for the cases n=1 and n=2.
PRODUCTION MATRIX
Define three arrays R, L, and S as
... R = superdiag[2,3,4,...]
... L = subdiag[1,2,3,...]
... S = diag[2,4,6,...]
with the indicated sequences on the main superdiagonal, the main subdiagonal and main diagonal, respectively, and 0's elsewhere. The array R+L is the production array for this triangle: the first row of (R+L)^n produces the n-th row of the triangle.
On the vector space of complex polynomials the array R, the raising operator, represents the operator p(x) - > d/dx (x^2*p(x)), and the array L, the lowering operator, represents the differential operator d/dx - see Formula (4) below.
The three arrays satisfy the commutation relations
... [R,L] = S, [R,S] = 2*R, [L,S] = -2*L
and hence give a representation of the Lie algebra sl(2).

			Table begins
  n\k|.....0.....1.....2.....3.....4.....5.....6
  ==============================================
  0..|.....1
  1..|.....0.....2
  2..|.....2.....0.....6
  3..|.....0....16.....0....24
  4..|....16.....0...120.....0...120
  5..|.....0...272.....0...960.....0...720
  6..|...272.....0..3696.....0..8400.....0..5040
Examples of recurrence relation
  T(4,2) = 3*(T(3,1) + T(3,3)) = 3*(16 + 24) = 120;
  T(6,4) = 5*(T(5,3) + T(5,5)) = 5*(960 + 720) = 8400.
Example of integral formula (6)
... Integral_{t = -1..1} (1-t^2)*(16-120*t^2+120*t^4)*(272-3696*t^2+8400*t^4-5040*t^6) dt = 2830336/1365 = -2^13*Bernoulli(12).
Examples of sign change statistic sc on snakes of type (0,0)
= = = = = = = = = = = = = = = = = = = = = =
.....Snakes....# sign changes sc.......t^sc
= = = = = = = = = = = = = = = = = = = = = =
n=1
...0 1 -2 0...........1................t
...0 2 -1 0...........1................t
yields R(1,t) = 2*t;
n=2
...0 1 -2 3 0.........2................t^2
...0 1 -3 2 0.........2................t^2
...0 2 1 3 0..........0................1
...0 2 -1 3 0.........2................t^2
...0 2 -3 1 0.........2................t^2
...0 3 1 2 0..........0................1
...0 3 -1 2 0.........2................t^2
...0 3 -2 1 0.........2................t^2
yields
R(2,t) = 2 + 6*t^2.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
K. Boyadzhiev, Derivative Polynomials for tanh, tan, sech and sec in Explicit Form, arXiv:0903.0117 [math.CA], 2009-2010.
M-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons, arXiv:math/0503175 [math.GM], 2005.
A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A (2007) 463, 2401-2414 doi:10.1098/rspa.2007.0001
Michael E. Hoffman, Derivative polynomials, Euler polynomials,and associated integer sequences, Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R21.
Michael E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly, 102 (1995), 23-30.
M. Josuat-Verges, Enumeration of snakes and cycle-alternating permutations, arXiv:1011.0929 [math.CO], 2010.
Shi-Mei Ma, Qi Fang, Toufik Mansour, Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374, 2021

Cf. A000182, A000364, A000828 (row sums), A008292, A008293, A008294, A104035, A155100.

R = proc(n) option remember;
if n=0 then RETURN(1);
else RETURN(expand(diff((u^2+1)*R(n-1), u))); fi;
end proc;
for n from 0 to 12 do
t1 := series(R(n), u, 20);
lprint(seriestolist(t1));
od:

Table[(-1)^(n + 1)*(-1)^((n - k)/2)*Sum[j!*StirlingS2[n + 1, j]*2^(n + 1 - j)*(-1)^(n + j - k)*Binomial[j - 1, k], {j, k + 1, n + 1}], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Jul 22 2017 *)

{T(n, k) = if( n<0 || k<0 || k>n, 0, if(n==k, n!, (k+1)*(T(n-1, k-1) + T(n-1, k+1))))};

{T(n, k) = my(A); if( n<0 || k>n, 0, A=1; for(i=1, n, A = ((1 + x^2) * A)'); polcoeff(A, k))}; /* Michael Somos, Jun 24 2017 */

GENERATING FUNCTION
E.g.f.:
(1)... F(t,z) = 1/(cos(z)-t*sin(z))^2 = Sum_{n>=0} R(n,t)*z^n/n! = 1 + (2*t)*z + (2+6*t^2)*z^2/2! + (16*t+24*t^3)*z^3/3! + ....
The e.g.f. equals the square of the e.g.f. of A104035.
Continued fraction representation for the o.g.f:
(2)... F(t,z) = 1/(1-2*t*z - 2*(1+t^2)*z^2/(1-4*t*z -...- n*(n+1)*(1+t^2)*z^2/(1-2*n*(n+1)*t*z -....
RECURRENCE RELATION
(3)... T(n,k) = (k+1)*(T(n-1,k-1) + T(n-1,k+1)).
ROW POLYNOMIALS
The polynomials R(n,t) satisfy the recurrence relation
(4)... R(n+1,t) = d/dt{(1+t^2)*R(n,t)} with R(0,t) = 1.
Let D be the derivative operator d/dt and U = t, the shift operator.
(5)... R(n,t) = (D + DUU)^n 1
RELATION WITH OTHER SEQUENCES
A) Derivative Polynomials A155100
The polynomials (1+t^2)*R(n,t) are the polynomials P_(n+2)(t) of A155100.
B) Bernoulli Numbers A000367 and A002445
Put S(n,t) = R(n,i*t), where i = sqrt(-1). We have the definite integral evaluation
(6)... Integral_{t = -1..1} (1-t^2)*S(m,t)*S(n,t) dt = (-1)^((m-n)/2)*2^(m+n+3)*Bernoulli(m+n+2).
The case m = n is equivalent to the result of [Grosset and Veselov]. The methods used there extend to the general case.
C) Zigzag Numbers A000111
(7)... R_n(1) = A000828(n+1) = 2^n*A000111(n+1).
D) Eulerian Numbers A008292
The polynomials R(n,t) are related to the Eulerian polynomials A(n,t) via
(8)... R(n,t) = (t+i)^n*A(n+1,(t-i)/(t+i))
with the inverse identity
(9)... A(n+1,t) = (-i/2)^n*(1-t)^n*R(n,i*(1+t)/(1-t)),
where {A(n,t)}n>=1 = [1,1+t,1+4*t+t^2,1+11*t+11*t^2+t^3,...] is the sequence of Eulerian polynomials and i = sqrt(-1).
E) Ordered set partitions A019538
(10)... R(n,t) = (-2*i)^n*T(n+1,x)/x,
where x = i/2*t - 1/2 and T(n,x) is the n-th row po1ynomial of A019538;
F) Miscellaneous
Column 1 is the sequence of tangent numbers - see A000182.
A000670(n+1) = (-i/2)^n*R(n,3*i).
A004123(n+2) = 2*(-i/2)^n*R(n,5*i).
A080795(n+1) =(-1)^n*(sqrt(-2))^n*R(n,sqrt(-2)). - Peter Bala, Aug 26 2011
From Leonid Bedratyuk, Aug 12 2012: (Start)
T(n,k) = (-1)^(n+1)*(-1)^((n-k)/2)*Sum_{j=k+1..n+1} j! *stirling2(n+1,j) *2^(n+1-j) *(-1)^(n+j-k) *binomial(j-1,k), see A059419.
Sum_{j=i+1..n+1}((1-(-1)^(j-i))/(2*(j-i))*(-1)^((n-j)/2)*T(n,j))=(n+1)*(-1)^((n-1-i)/2)*T(n-1,i), for n>1 and  0G.f.:  1/G(0,t,x), where G(k,t,x) = 1 - 2*t*x - 2*k*t*x - (1+t^2)*(k+2)*(k+1)*x^2/G(k+1,t,x); (continued fraction due to T. J. Stieltjes). - Sergei N. Gladkovskii, Dec 27 2013

A003245 Nearest integer to -4n/Bernoulli(2n).

Original entry on oeis.org

0, -24, 240, -504, 480, -264, 95, -24, 5, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

sign

Douglas C. Ravenel, Complex cobordism theory for number theorists, Lecture Notes in Mathematics, 1326, Springer-Verlag, Berlin-New York, 1988, pp. 123-133.
F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg, 2nd ed. 1994, p. 130.

Cf. A000367/A002445.

Table[Round[(-4n)/BernoulliB[2n]], {n, 0, 75}] (* Alonso del Arte, Jul 19 2012 *)

A075180 Denominators from e.g.f. 1/(1-exp(-x)) - 1/x.

Original entry on oeis.org

2, 12, 1, 120, 1, 252, 1, 240, 1, 132, 1, 32760, 1, 12, 1, 8160, 1, 14364, 1, 6600, 1, 276, 1, 65520, 1, 12, 1, 3480, 1, 85932, 1, 16320, 1, 12, 1, 69090840, 1, 12, 1, 541200, 1, 75852, 1, 2760, 1, 564, 1, 2227680, 1, 132, 1, 6360, 1, 43092, 1, 6960, 1, 708, 1, 3407203800, 1, 12, 1, 32640, 1, 388332, 1, 120, 1, 9372, 1, 10087262640, 1, 12

Offset: 0

Wolfdieter Lang, Sep 06 2002

Denominators of -zeta(-n), n >= 0, where zeta is Riemann's zeta function.

Numerators are +1, A060054(n+1), n >= 1.

			1/2, 1/12, 0, -1/120, 0, 1/252, 0, -1/240, 0, 1/132, 0, -691/32760, ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 807, combined eqs. 23.2.11,14 and 15.

Antti Karttunen, Table of n, a(n) for n = 0..16384
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 807, combined eqs. 23.2.11,14 and 15.

Cf. A060054, A006232 with A006233.

Cf. A006953, A006863, A079612, A242179, A106831, A000142, A300711.

a075180 n = a075180_list !! n
a075180_list = map (denominator . sum) $ zipWith (zipWith (%))
   (zipWith (map . (*)) a000142_list a242179_tabf) a106831_tabf
-- Reinhard Zumkeller, Jul 04 2014

a := n -> denom(bernoulli(n+1,1)/(n+1)); # Peter Luschny, Apr 22 2009

a[m_] := Sum[(-2)^(-k-1) k! StirlingS2[m,k],{k,0,m}]/(2^(m+1)-1); Table[Denominator[a[i]], {i,0,20}] (* Peter Luschny, Apr 29 2009 *)
Table[Denominator[Zeta[-n]], {n, 0, 49}] (* Alonso del Arte, Jan 13 2012 *)
CoefficientList[ Series[ EulerGamma - HarmonicNumber[n] + Log[n], {n, Infinity, 48}], 1/n] // Rest // Denominator (* Jean-François Alcover, Mar 28 2013 *)
With[{nn=50},Denominator[CoefficientList[Series[1/(1-Exp[-x])-1/x,{x,0,nn}],x] Range[0,nn-1]!]] (* Harvey P. Dale, Apr 13 2016 *)

x='x+O('x^66);
egf = 1/(1-exp(-x)) - 1/x;
v=Vec(serlaplace(egf));
vector(#v,n, denominator(v[n]))
/* Joerg Arndt, Mar 28 2013 */

A075180(n) = denominator(bernfrac(n+1)/(n+1)); \\ Antti Karttunen, Dec 19 2018, after Maple-program.

a(n) = denominator(-Zeta(-n)) = denominator(((-1)^(n+1))*B(n+1)/(n+1)), n >= 0, with Riemann's zeta function and the Bernoulli numbers B(n).
a(n) = denominators from e.g.f. (B(-x) - 1)/x, with B(x) = x/(exp(x) - 1), e.g.f. for Bernoulli numbers A027641(n)/A027642(n), n >= 0.
From Jianing Song, Apr 05 2021: (Start)
a(2n-1) = A006863(n)/2 for n > 0. By the comments in A006863, A006863(n) = A079612(2n) for n > 0. Hence a(n) = A079612(n+1)/2 all odd n. For all even n > 0, we have a(n) = 1, which is also equal to A079612(n+1)/2.
For odd n, a(n) is the product of p^(e+1) where p^e*(p-1) divides n+1 but p^(e+1)*(p-1) does not. For example, a(11) = 2^3 * 3^2 * 5^1 * 7^1 * 13^1 = 32760.
a(2n-1) = A002445(n)*(2n)/A300711(n), n > 0. (End)
a(2*n-1) = A006953(n) for n >= 1. - Georg Fischer, Dec 01 2022

More terms from Antti Karttunen, Dec 19 2018

A117972 Numerator of zeta'(-2n), n >= 0.

Original entry on oeis.org

1, -1, 3, -45, 315, -14175, 467775, -42567525, 638512875, -97692469875, 9280784638125, -2143861251406875, 147926426347074375, -48076088562799171875, 9086380738369043484375, -3952575621190533915703125

Offset: 0

Eric W. Weisstein, Apr 06 2006

In A160464 the coefficients of the ES1 matrix are defined. This matrix led to the discovery that the successive differences of the ES1[1-2*m,n] coefficients for m = 1, 2, 3, ..., are equal to the values of zeta'(-2n), see also A094665 and A160468. - Johannes W. Meijer, May 24 2009
A048896(n), n >= 1: Numerators of Maclaurin series for 1 - ((sin x)/x)^2,
  a(n), n >= 2: Denominators of Maclaurin series for 1 - ((sin x)/x)^2, the correlation function in Montgomery's pair correlation conjecture. - Daniel Forgues, Oct 16 2011
From Andrey Zabolotskiy, Sep 23 2021: (Start)
zeta'(-2n), which is mentioned in the Name, is irrational. For n > 0, a(n) is the numerator of the rational fraction g(n) = Pi^(2n)*zeta'(-2n)/zeta(2n+1). The denominator is 4*A048896(n-1). g(n) = f(n) for n > 0, where f(n) is given in the Formula section. Also, f(n) = Bernoulli(2n)/z(n)/4 (see Formula section) for all n.
For n = 0, zeta'(0) = -log(2Pi)/2, g(0) can be set to 0 because of the infinite denominator. However, a(0) is set to 1 because it is the numerator of f(0).
It seems that -4*f(n)*alpha_n = A000182(n), where alpha_n = A191657(n, p(n)) / A191658(n, p(n)) [where p(n) = A000041(n)] is the n-th "elementary coefficient" from the paper by Izaurieta et al. (End)

			-1/4, 3/4, -45/8, 315/4, -14175/8, 467775/8, -42567525/16, ...
-zeta(3)/(4*Pi^2), (3*zeta(5))/(4*Pi^4), (-45*zeta(7))/(8*Pi^6), (315*zeta(9))/(4*Pi^8), (-14175*zeta(11))/(8*Pi^10), ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Fernando Izaurieta, Ricardo Ramírez and Eduardo Rodríguez, Dirac Matrices for Chern-Simons Gravity, arXiv:1106.1648 [math-ph], 2011-2012.
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function

Cf. A000367, A002445, A049606, A046988, A002432, A117973, A048896.
From Johannes W. Meijer, May 24 2009: (Start)
Cf. A160464, A094665 and A160468.
Absolute values equal row sums of A160468. (End)
Cf. A191657, A191658, A000182, A000041.

# Without rational arithmetic
a := n -> (-1)^n*(2*n)!*2^(add(i,i=convert(n,base,2))-2*n);
# Peter Luschny, May 02 2009

Table[Numerator[(2 n)!/2^(2 n + 1) (-1)^n], {n, 0, 30}]

L:taylor(1/x*sin(sqrt(x))^2,x,0,15); makelist(denom(coeff(L,x,n))*(-1)^(n+1),n,0,15); /* Vladimir Kruchinin, May 30 2011 */

a(n) = numerator(f(n)) where f(n) = (2*n)!/2^(2*n + 1)(-1)^n, from the Mathematica code.
From Terry D. Grant, May 28 2017: (Start)
|a(n)| = A049606(2n).
a(n) = -numerator(Bernoulli(2n)/z(n)) where Bernoulli(2n) = A000367(n) / A002445(n) and z(n) = A046988(n) / A002432(n) for n > 0. (End) [Corrected by Andrey Zabolotskiy, Sep 23 2021]

First term added, offset changed and edited by Johannes W. Meijer, May 15 2009

A120080 Numerators of expansion of original Debye function D(3,x).

Original entry on oeis.org

1, -3, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -691, 0, 1, 0, -3617, 0, 43867, 0, -174611, 0, 77683, 0, -236364091, 0, 657931, 0, -3392780147, 0, 1723168255201, 0, -7709321041217, 0, 151628697551, 0, -26315271553053477373, 0, 154210205991661, 0, -261082718496449122051

Offset: 0

Wolfdieter Lang, Jul 20 2006

Denominators are given in A120081.

See the W. Lang link below for more details on the general case D(n,x), n= 1, 2, ... D(3,x) is the e.g.f. of the rational sequence {3*B(n)/(n+3)}, n >= 0. See A227570/A227571.

			Rationals r(n): [1, -3/8, 1/20, 0, -1/1680, 0, 1/90720, 0, ...].

L. D. Landau, E. M. Lifschitz: Lehrbuch der Theoretischen Physik, Band V: Statistische Physik, Akademie Verlag, Leipzig, p. 195, equ. (63.5) and footnote 1 on p. 197.

G. C. Greubel, Table of n, a(n) for n = 0..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 998, equ. 27.1.1 for n=3, with a factor (x^3)/3 extracted.
Wolfdieter Lang, Rationals r(n), and general remarks on the e.g.f. D(n,x).

Cf. A000367, A002445, A027641, A027642, A120081, A227570, A227571.

[Numerator(3*Bernoulli(n)/((n+3)*Factorial(n))): n in [0..50]]; // G. C. Greubel, May 01 2023

max = 39; Numerator[CoefficientList[Integrate[Normal[Series[(3*(t^3/(Exp[t] - 1)))/x^3, {t, 0, max}]], {t, 0, x}], x]] (* Jean-François Alcover, Oct 04 2011 *)
Table[Numerator[3*BernoulliB[n]/((n+3)*n!)], {n,0,50}] (* G. C. Greubel, May 01 2023 *)

def A120080(n): return numerator(3*bernoulli(n)/((n+3)*factorial(n)))
[A120080(n) for n in range(51)] # G. C. Greubel, May 01 2023

D(x) = D(3,x) := (3/x^3)*Integral_{0..x} t^3/(exp(t)-1) dt.
a(n) = numerator(r(n)), with r(n) = [x^n]( 1 - 3*x/8 + Sum_{k >= 1} (3*B(2*k)/((2*k+3)*(2*k)!))*x^(2*k) ) (in lowest terms), |x| < 2*pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).
a(n) = numerator(3*B(n)/((n+3)*n!)), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n). See the comment on the e.g.f. D(3,x) above. - Wolfdieter Lang, Jul 16 2013

A027762 Denominator of Sum_{p prime, p-1 divides 2*n} 1/p.

Original entry on oeis.org

6, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, 1806, 690, 282, 46410, 66, 1590, 798, 870, 354, 56786730, 6, 510, 64722, 30, 4686, 140100870, 6, 30, 3318, 230010, 498, 3404310, 6, 61410, 272118, 1410, 6, 4501770

Offset: 1

N. J. A. Sloane

From the von Staudt-Clausen theorem, denominator(B_2n) = product of primes p such that (p-1)|2n.

Same as A002445.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Th. 118.
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
Index entries for sequences related to Bernoulli numbers.

Cf. A027761, A006954.

Cf. A027760.

a(n)=
{
    my(s=0);
    forprime (p=2, 2*n+1, if( (2*n)%(p-1)==0, s+=1/p ) );
    return( denominator(s) );
}
/* Joerg Arndt, May 06 2012 */

a(n) = A002445(n). [Joerg Arndt, May 06 2012]

a(n) = A027760(2*n). - Ridouane Oudra, Feb 22 2022

A080092 Irregular triangle read by rows, giving prime sequences (p-1|2n) appearing in the n-th von Staudt-Clausen sum.

Original entry on oeis.org

2, 2, 3, 2, 3, 5, 2, 3, 7, 2, 3, 5, 2, 3, 11, 2, 3, 5, 7, 13, 2, 3, 2, 3, 5, 17, 2, 3, 7, 19, 2, 3, 5, 11, 2, 3, 23, 2, 3, 5, 7, 13, 2, 3, 2, 3, 5, 29, 2, 3, 7, 11, 31, 2, 3, 5, 17, 2, 3, 2, 3, 5, 7, 13, 19, 37, 2, 3, 2, 3, 5, 11, 41, 2, 3, 7, 43, 2, 3, 5, 23, 2, 3, 47, 2, 3, 5, 7, 13, 17, 2, 3

Offset: 1

Eric W. Weisstein, Jan 27 2003

From Gary W. Adamson & Mats Granvik, Aug 09 2008: (Start)
The von Staudt-Clausen theorem has two parts: generating denominators of the B_2n and the actual values. Both operations can be demonstrated in triangles A143343 and A080092 by following the procedures outlined in [Wikipedia - Bernoulli numbers] and summarized in A143343.
A046886(n-1) = number of terms in row n.
The same terms in A143343 may be extracted from triangle A138239.
Extract primes from even numbered rows of triangle A143343 but also include "2" as row 1. The rows are thus 1, 2, 4, 6, ..., generating denominators of B_1, B_2, B_4, ..., as well as B_1, B_2, B_4, ..., as two parts of the von Staudt-Clausen theorem.
The denominator of B_12 = 2730 = 2*3*5*7*13 = A027642(12) and A002445(6).
For example, B_12 = -691/2730 = (1 - 1/2 - 1/3 - 1/5 - 1/7 - 1/13).
The second operation is the von Staudt-Clausen representation of Bn, obtained by starting with "1" and then subtracting the reciprocals of terms in each row. (Cf. A143343 for a detailed explanation of the operations.) (End)

			First few rows of the triangle:
  2;
  2, 3;
  2, 3, 5;
  2, 3, 7;
  2, 3, 5;
  2, 3, 11;
  2, 3, 5, 7, 13;
  2, 3;
  ...
Sum for n=1 is 1/2 + 1/3, so terms are 2, 3;
sum for n=2 is 1/2 + 1/3 + 1/5, so terms are 2, 3, 5; etc.

Eric Weisstein's World of Mathematics, von Staudt-Clausen Theorem.
Wikipedia, Von Staudt-Clausen theorem.

Cf. A000146, A002445, A027642, A138239, A143343.

row[n_] := Select[ Prime /@ Range[n+1], Divisible[2n, # - 1] &]; Flatten[Table[row[n], {n, 0, 25}]] (* Jean-François Alcover, Oct 12 2011 *)

Edited by N. J. A. Sloane, Nov 01 2009 at the suggestion of R. J. Mathar

A176289 Denominators of the rational sequence with e.g.f. (x/2)*(1+exp(-x))/(1-exp(-x)).

Original entry on oeis.org

1, 1, 6, 1, 30, 1, 42, 1, 30, 1, 66, 1, 2730, 1, 6, 1, 510, 1, 798, 1, 330, 1, 138, 1, 2730, 1, 6, 1, 870, 1, 14322, 1, 510, 1, 6, 1, 1919190, 1, 6, 1, 13530, 1, 1806, 1, 690, 1, 282, 1, 46410, 1, 66, 1, 1590, 1, 798, 1, 870, 1, 354, 1, 56786730, 1, 6, 1, 510

Offset: 0

Paul Curtz, Apr 14 2010

Denominator of the Bernoulli number B_n, except a(1)=1. A minor variant of the Bernoulli denominators A027642.
The sequence of fractions A164555(n)/A027642(n) = 1/1, 1/2, 1/6, 0/1, -1/30, ...
and the sequence of fractions A027641(n)/A027642(n) = B_n = 1/1, -1/2, 1/6, 0/1, -1/30, ... differ only (by a sign) at n=1. The arithmetic mean of both sequences is 1/1, 0/1, 1/6, 0/1, -1/30, ..., equal to the aerated sequence A000367(n)/A002445(n). The definition here provides the denominators of this sequence of arithmetic means.

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A027641, A027642, A164555, A176327 (numerators), A141056.

seq(denom((bernoulli(i,0)+bernoulli(i,1))/2),i=0..64); # Peter Luschny, Jun 17 2012

Join[{1,1},Rest[Denominator[BernoulliB[Range[80]]]]] (* Harvey P. Dale, Jun 18 2012 *)

apply(deniominator, Vec(serlaplace((x/2)*(1+exp(-x))/(1-exp(-x))))) \\ Charles R Greathouse IV, Sep 26 2017

A176289(n) = if(1==n,n,denominator(bernfrac(n))); \\ Antti Karttunen, Dec 19 2018

a(2*n) = A002445(n), a(2*n+1)=1.

a(n) = A027642(n) for n <> 1.

More terms from Harvey P. Dale, May 03 2012

New name from Peter Luschny, Jun 18 2012

A004193 a(n) = -(-1)^n2(2n+1)!Bernoulli(2n)/(n!2^n).

Original entry on oeis.org

1, 1, 5, 63, 1575, 68409, 4729725, 488783295, 71982456975, 14550187083705, 3916321542458325, 1368981608178405375, 608576219802039864375, 337967570725260384533625, 230885276313275432674678125, 191452972504088518574149173375, 190442238700388913304502070009375

Offset: 1

David W. Cantrell (DWCantrell(AT)sigmaxi.net)

J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere, NY, 1987, p. 35, Eq. 4:2:1.

G. C. Greubel, Table of n, a(n) for n = 1..100
Index entries for sequences related to Bernoulli numbers.

Bernoulli numbers at even indices are A000367/A002445.

a:= n-> -(-1)^n*2*(2*n+1)!*bernoulli(2*n)/(n!*2^n):
seq(a(n), n=1..20);  # Alois P. Heinz, Jun 13 2016

Table[-((-1)^n 2(2n+1)!BernoulliB[2n])/(n! 2^n),{n,20}] (* Harvey P. Dale, Oct 05 2012 *)
Table[2 (2n+1)!! Abs@BernoulliB[2n], {n, 20}] (* Vladimir Reshetnikov, Jun 05 2016 *)

a(n)=if(n<1,0,-(-1)^n*2*(2*n+1)!*bernfrac(2*n)/(n!*2^n))

a(n) ~ 16 * 2^(n+1/2) * Pi^(1/2-2*n) * n^(3/2) * (n/e)^(3*n). - Vladimir Reshetnikov, Sep 05 2016
From Peter Luschny, May 17 2018: (Start)
a(n) ~ 8*sqrt(2*n*Pi)*(2*Pi)^n*(n/(Pi*e))^(3*n)*(2*n+1).
a(n) = |2^(n+2)*Pochhammer(1/2, n+1)*Bernoulli(2*n)|. (End)
a(n) = -(-2)^(n+3)*n*Zeta(1-2*n)*(n+1/2)!/sqrt(Pi). - Peter Luschny, Jun 21 2020

A033469 Denominator of Bernoulli(2n,1/2).

Original entry on oeis.org

1, 12, 240, 1344, 3840, 33792, 5591040, 49152, 16711680, 104595456, 173015040, 289406976, 22900899840, 201326592, 116769423360, 7689065201664, 1095216660480, 51539607552, 65942866278481920, 824633720832, 7438196161904640, 3971435999526912

Offset: 0

N. J. A. Sloane

nonn

From the von Staudt-Clausen theorem it follows that a(n) can be computed without using Bernoulli polynomials or the 'denominator'-function (see the Sage implementation). - Peter Luschny, Mar 24 2014

J. R. Philip, The symmetrical Euler-Maclaurin summation formula, Math. Sci., 6, 1981, pp. 35-41.

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, von Staudt-Clausen Theorem.
Index entries for sequences related to Bernoulli numbers.

Cf. A001896.

with(numtheory); seq(denom(bernoulli(2*n, 1/2)), n=0..20);

Table[ BernoulliB[2*n, 1/2] // Denominator, {n, 0, 18}] (* Jean-François Alcover, Apr 15 2013 *)
a[ n_] := If[ n < 0, 0, (2 n)! SeriesCoefficient[ x/2 / Sinh[x/2], {x, 0, 2 n}] // Denominator]; (* Michael Somos, Sep 21 2016 *)

a(n)=denominator(subst(bernpol(2*n,x),x,1/2)); \\ Joerg Arndt, Apr 17 2013

def A033469(n):
    if n == 0: return 1
    M = map(lambda i: i+1, divisors(2*n))
    return 2^(2*n-1)*mul(filter(lambda s: is_prime(s), M))
[A033469(n) for n in (0..21)] # Peter Luschny, Mar 24 2014

a(n) = denominator(2*(2*Pi)^(-2*n)*(2*n)!*Li_{2*n}(-1)). - Peter Luschny, Jun 29 2012
a(n) = A081294(n) * A002445(n) for n > 0. - Paul Curtz, Apr 17 2013
Apparently, denominators of the fractions with e.g.f. (x/2) / sinh(x/2). - Tom Copeland, Sep 17 2016

More terms from Joerg Arndt, Apr 17 2013

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