A132092 -id:A132092 - cp's OEIS frontend

A001896 Numerators of cosecant numbers -2(2^(2n - 1) - 1)Bernoulli(2n); also of Bernoulli(2n, 1/2) and Bernoulli(2n, 1/4).

1, -1, 7, -31, 127, -2555, 1414477, -57337, 118518239, -5749691557, 91546277357, -1792042792463, 1982765468311237, -286994504449393, 3187598676787461083, -4625594554880206790555, 16555640865486520478399, -22142170099387402072897

Offset: 0

N. J. A. Sloane

|A001896(n)|*Pi^(2n)/A001897(n) is the value of the multi zeta function z(2,2,...,2) with n 2's, where z(k_l,k_2,...,k_n) = Sum_{i_n >= i_(n-1) >= ... >= i_1 >= 1}1/((i_1)^k_1 (i_2)^k_2 ... (i_n)^k_n). The proof is simple: start with the product expansion sin(Pi x)/(Pi x) = Product_{r>=1}(1-x^2/r^2), take reciprocals, and expand the right side. The coefficient of x^(2n) is seen to be z(2,2,...,2) with n 2's. - David Callan, Aug 27 2014

See A062715 for a method of obtaining the cosecant numbers from the square of Pascal's triangle. - Peter Bala, Jul 18 2013

			1, -1/12, 7/240, -31/1344, 127/3840, -2555/33792, 1414477/5591040, -57337/49152, 118518239/16711680, ... = a(n)/A033469(n).
Cosecant numbers {-2*(2^(2*n-1)-1)*Bernoulli(2*n)} are 1, -1/3, 7/15, -31/21, 127/15, -2555/33, 1414477/1365, -57337/3, 118518239/255, -5749691557/399, 91546277357/165, -1792042792463/69, 1982765468311237/1365, -286994504449393/3, 3187598676787461083/435, ... = a(n)/A001897(n).

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 187.
S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. Math. 46 (1914), 33-51.
N. E. Nörlund, Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin, 1924, p. 458.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. See Table 3.3.
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).

Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008; Page 7, 3rd table, (B^sin)_1,n is identical to |A001896| / A001897.
S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. Math. 46 (1914), 33-51. [Annotated scanned copy of pages 38-51 only, plus notes]
Masanobu Kaneko, Maneka Pallewatta, and Hirofumi Tsumura, On Polycosecant Numbers, J. Integer Seq. 23 (2020), no. 6, 17 pp. See line k=1 of Table 1 p. 3.
D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]
Index entries for sequences related to Bernoulli numbers.

Cf. A001897 (denominators), A033469, A036280, A062715, A145901.

Cf. A132092-A132099.

seq(numer(bernoulli(2*n, 1/2)), n=0..20);

a[n_] := -2*(2^(2*n-1)-1)*BernoulliB[2*n]; Table[a[n], {n, 0, 20}] // Numerator (* Jean-François Alcover, Sep 11 2013 *)

a(n) = numerator(-2*(2^(2*n-1)-1)*bernfrac(2*n)); \\ Michel Marcus, Mar 01 2015

def A001896_list(len):
    R, C = [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        for k in range(n, 0, -1):
            C[k] = C[k-1] / (8*k*(2*k+1))
        C[0] = -sum(C[k] for k in (1..n))
        R.append((C[0]*factorial(2*n)).numerator())
    return R
A001896_list(18) # Peter Luschny, Feb 20 2016

a(n) = numerator((-Pi^2)^(-n)*Integral_{x=0..1} (log(x/(1-x)))^2*n). - Groux Roland, Nov 10 2009
a(n) = numerator((-1)^(n+1)*(2*Pi)^(-2*n)*(2*n)!*Li_{2*n}(-1)). - Peter Luschny, Jun 29 2012
E.g.f. 2*x*exp(x)/(exp(2*x) - 1) = 1 - 1/3*x^2/2! + 7/15*x^4/4! - 31/21*x^6/6! + .... = Sum_{n >= 0} a(n)/A001897(n)*x^(2*n)/(2*n)!. - Peter Bala, Jul 18 2013
a(n) = numerator((-1)^n*I(n)), where I(n) = 2*Pi*Integral_{z=-oo..oo} (z^n / (exp(-Pi*z) + exp(Pi*z)))^2. - Peter Luschny, Jul 25 2021

A001897 Denominators of cosecant numbers: -2(2^(2n-1)-1)Bernoulli(2n).

Original entry on oeis.org

1, 3, 15, 21, 15, 33, 1365, 3, 255, 399, 165, 69, 1365, 3, 435, 7161, 255, 3, 959595, 3, 6765, 903, 345, 141, 23205, 33, 795, 399, 435, 177, 28393365, 3, 255, 32361, 15, 2343, 70050435, 3, 15, 1659, 115005, 249, 1702155, 3, 30705, 136059, 705, 3, 2250885, 3, 16665, 2163

Offset: 0

N. J. A. Sloane

Same as half the denominators of the even-indexed Bernoulli numbers B_{2*n} for n>0, by the von Staudt-Clausen theorem and Fermat's little theorem. - Bernd C. Kellner and Jonathan Sondow, Jan 02 2017 [This is implemented in the second Maple program. - Peter Luschny, Aug 21 2021]

			Cosecant numbers {-2*(2^(2*n-1)-1)*Bernoulli(2*n)} are 1, -1/3, 7/15, -31/21, 127/15, -2555/33, 1414477/1365, -57337/3, 118518239/255, -5749691557/399, 91546277357/165, -1792042792463/69, 1982765468311237/1365, -286994504449393/3, 3187598676787461083/435, ... = A001896/A001897.

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 187.
S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. Math. 46 (1914), 33-51.
N. E. Nörlund, Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin, 1924, p. 458.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. See Table 3.3.
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).

Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, Page 7, 3rd table, (B^sin)_1,n is identical to |A001896| / A001897.
S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. Math. 46 (1914), 33-51. [Annotated scanned copy of pages 38-51 only, plus notes]
Masanobu Kaneko, Maneka Pallewatta, and Hirofumi Tsumura, On Polycosecant Numbers, J. Integer Seq. 23 (2020), no. 6, 17 pp. See line k=1 of Table 1 p. 3.
D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer 1924, p. 27.
N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]

Cf. A001896 (numerators), A027642, A033469, A160014, A241171, A277087, A340556.

Cf. A132092-A132099,

[Denominator(2*(1-2^(2*n-1))*Bernoulli(2*n)): n in [0..55]]; // G. C. Greubel, Apr 06 2019

b := n -> bernoulli(n)*2^add(i,i=convert(n,base,2));
a := n -> denom(b(2*n)); # Peter Luschny, May 02 2009
# Alternative :
Clausen := proc(n) local i,S; map(i->i+1, numtheory[divisors](n));
S := select(isprime, %); if S <> {} then mul(i,i=S) else NULL fi end:
A001897_list := n -> [1,seq(Clausen(2*i)/2,i=1..n-1)];
A001897_list(52); # Peter Luschny, Oct 03 2011

a[n_] := Denominator[-2*(2^(2*n-1)-1)*BernoulliB[2*n]]; Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Sep 11 2013 *)

a(n) = denominator(-2*(2^(2*n-1)-1)*bernfrac(2*n)); \\ Michel Marcus, Apr 06 2019

def A001897(n):
    if n == 0:
        return 1
    M = (d + 1 for d in divisors(2 * n))
    return prod(s for s in M if is_prime(s)) / 2
[A001897(n) for n in range(55)]  # Peter Luschny, Feb 20 2016

a(0)=1, a(n)=(1/2)*A002445(n) for n>=1. - Joerg Arndt, May 07 2012
a(n) = denominator((2*n)!*Li_{2*n}(1)) for n > 0. - Peter Luschny, Jun 29 2012
a(0)=1, a(n) = (1/2)*A027642(2*n) = (3/2)*A277087(n) for n>=1. - Jonathan Sondow, Dec 14 2016
From Peter Luschny, Sep 06 2017: (Start)
a(n) = denominator(r(n)) where r(n) = Sum_{0..n} (-1)^(n-k)*A241171(n, k)/(2*k+1).
a(n) = denominator(bernoulli(2*n, 1/2))/4^n = A033469(n)/4^n. (End)
Apparently a(n) = denominator(Sum_{k=0..2*n-2} (-1)^k*E2(2*n-1, k+1)/binomial(4*n-1, k+1)), where E2(n, k) denotes the second-order Eulerian numbers A340556. - Peter Luschny, Feb 17 2021

A132099 Denominators of Blandin-Diaz compositional Bernoulli numbers (B^Z^2)_1,n.

Original entry on oeis.org

1, 8, 432, 144, 324000, 64800, 16669800

Offset: 0

Jonathan Vos Post, Aug 09 2007

			1, -1/8, 11/432, 1/144, -217/324000, -157/64800, -21503/16669800.

Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, p. 9, 2nd table.

Numerators are A132098.

Cf. A132092-A132097.

A132094 Numerators of expansion of e.g.f. x^2/(2*(cos(x)-1)), even powers only.

Original entry on oeis.org

-1, -1, -1, -5, -7, -15, -7601, -91, -3617, -745739, -3317609, -5981591, -5436374093, -213827575, -213745149261, -249859397004145, -238988952277727, -28354566442037, -26315271553053477373, -108409774812137683, -3394075340453838586663, -62324003400640902910331

Offset: 1

Jonathan Vos Post, Aug 09 2007

Numerators and denominators given only for even n (odd n have numerators = 0).

			-1, 0, -1/6, 0, -1/10, 0, -5/42, 0, -7/30, 0, -15/22, 0, -7601/2730, 0.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. See Table 3.3.

Robert Israel, Table of n, a(n) for n = 1..288
Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, p. 8, 2nd table.

Denominators are A132095.

Cf. A132092-A132099.

A132094 := proc(n) add( 2*(-1)^i*x^(2*i)/(2*i+2)!,i=0..n+1) ; numer(coeftayl(-1/%,x=0,n)*n!) ; end: for n from 0 to 46 by 2 do printf("%d, ",A132094(n)) ; od: # R. J. Mathar, Oct 18 2007

A132094[n_] := (s = Sum[ 2*(-1)^i*x^(2*i)/(2*i + 2)!, {i, 0, n + 1}]; Numerator[SeriesCoefficient[-1/s, {x, 0, n}]*n!]);
Table[A132094[n], {n, 0, 46, 2}] (* Jean-François Alcover, Nov 24 2017, after R. J. Mathar *)

my(x='x+O('x^50), v=apply(numerator, Vec(serlaplace(x^2/(2*(cos(x)-1)))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Jan 25 2024

Asymptotic series 2*Psi(1,x) + x*Psi(2,x) ~ Sum_{n>=1} (-1)^n* a(n)/(A132095(n)*x^(2*n-1)) as x -> oo. - Robert Israel, May 27 2015

More terms from R. J. Mathar, Oct 18 2007

Meaningful name from Joerg Arndt, Jan 25 2024

A132095 Denominators of expansion of e.g.f. x^2/(2*(cos(x)-1)), even powers only.

Original entry on oeis.org

1, 6, 10, 42, 30, 22, 2730, 6, 34, 798, 330, 46, 2730, 6, 290, 14322, 510, 2, 54834, 6, 4510, 1806, 690, 94, 46410, 66, 530, 798, 174, 118, 56786730, 6, 170, 64722, 30, 1562, 140100870, 6, 2, 474, 230010, 166, 3404310, 6, 20470, 272118, 1410, 2, 900354, 6

Offset: 1

Jonathan Vos Post, Aug 09 2007

Numerators and denominators given only for even n (odd n have numerators = 0).

			-1, 0, -1/6, 0, -1/10, 0, -5/42, 0, -7/30, 0, -15/22, 0, -7601/2730, 0.

Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers , Afr. Diaspora J. Math., Volume 7, Number 2 (2008).
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. See Table 3.3.

Robert Israel, Table of n, a(n) for n = 1..1000
Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, p. 8, 2nd table.

Numerators are A132094.

Cf. A132092-A132099.

A132095 := proc(n) add( 2*(-1)^i*x^(2*i)/(2*i+2)!,i=0..n/2+1) ; denom(coeftayl(-1/%,x=0,n)*n!) ; end: for n from 0 to 46 by 2 do printf("%d, ",A132095(n)) ; od: # R. J. Mathar, Oct 18 2007

A132095[n_] := (s = Sum[ 2*(-1)^i*x^(2*i)/(2*i + 2)!, {i, 0, n/2 + 1}] ; Denominator[SeriesCoefficient[-1/s, {x, 0, n}]*n!]) ;
Table[ A132095[n], {n, 0, 100, 2}] (* Jean-François Alcover, Nov 24 2017, after R. J. Mathar *)

my(x='x+O('x^100), v=apply(denominator, Vec(serlaplace(x^2/(2*(cos(x)-1)))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Jan 25 2024

Asymptotic series 2*Psi(1,x) + x*Psi(2,x) ~ Sum(n>=1, (-1)^n* A132094(n)/(a(n)*x^(2*n-1)) as x -> infinity. - Robert Israel, May 27 2015

More terms from R. J. Mathar, Oct 18 2007 and Oct 20 2009

Meaningful name from Joerg Arndt, Jan 25 2024

A132097 Denominators of Blandin-Diaz compositional Bernoulli numbers (B^Z)_1,n.

Original entry on oeis.org

1, 4, 72, 96, 21600, 640, 5080320, 580608, 326592000, 20736000, 2529128448, 1094860800, 1298164008960000, 399435079680000, 11298306539520000, 231760134144000, 48978158848819200000, 768284844687360000, 81541143706048266240000, 1009797445276139520000, 467359502609929273344000000

Offset: 0

Jonathan Vos Post, Aug 09 2007

			1, -1/4, 1/72, 1/96, 61/21600, -1/640, -12491/5080320, -479/680608.

Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, p. 9, 1st table.

Numerators are A132096.

Cf. A132092-A132099.

nn = 21; A = Inverse[Table[Table[If[n >= k, Binomial[n - 1, k - 1]/(n - k + 1)^2, 0], {k, 1, nn}], {n, 1, nn}]]; Denominator[A[[All, 1]]] (* Mats Granvik, Feb 03 2018 *)

a(n) = denominator(f(n)), where f(0) = 1, f(n) = -Sum_{k=0..n-1} f(k) * binomial(n,k) / (n-k+1)^2. - Daniel Suteu, Feb 23 2018

A132098 Numerators of Blandin-Diaz compositional Bernoulli numbers (B^Z^2)_1,n.

Original entry on oeis.org

1, -1, 11, 1, -217, -157, -21503

Offset: 0

Jonathan Vos Post, Aug 09 2007

			1, -1/8, 11/432, 1/144, -217/324000, -157/64800, -21503/16669800.

Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, p. 9, 2nd table.

Denominators are A132099.

Cf. A132092-A132098.

A006568 Denominators of generalized Bernoulli numbers.

Original entry on oeis.org

1, 3, 18, 90, 270, 1134, 5670, 2430, 7290, 133650, 112266, 1990170, 9950850, 2296350, 984150, 117113850, 351341550, 33657930, 21597171750, 3410079750, 572893398, 33613643250, 834229509750, 108812544750, 544062723750, 18280507518, 105464466450, 18690647109750

Offset: 0

N. J. A. Sloane

Triangle A209518 * [1, -1/3, 1/18, 1/90, ...] = [1, 0, 0, 0, 0, ...]. - Gary W. Adamson, Mar 09 2012

			a(0), a(1), a(2), ... = (1, -1/3, 1/18, ...) = leftmost column of the inverse of the 3 X 3 matrix [1; 1, 3; 1, 4, 6; ...].

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Berhanu, Hunduma Legesse, Arithmetical properties of hypergeometric bernoulli numbers, Indagationes Mathematicae, 2016.
Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, Page 7, 2nd table is identical to A006569/A006568.
Abdul Hassen and Hieu D. Nguyen, Hypergeometric Zeta Functions, arXiv:math/0509637 [math.NT], Sep 27 2005.
F. T. Howard, A sequence of numbers related to the exponential function, Duke Math. J., 34 (1967), 599-615.
Index entries for sequences related to Bernoulli numbers.

Cf. A006569, A132092-A132099, A209518.

rows = 28; M = Table[If[n-1 <= k <= n, 0, Binomial[n, k]], {n, 2, rows+1}, {k, 0, rows-1}] // Inverse;
M[[All, 1]] // Denominator (* Jean-François Alcover, Jul 14 2018 *)

def A006568_list(len):
    f, R, C = 1, [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        f *= n
        for k in range(n, 0, -1):
            C[k] = C[k-1] / (k+2)
        C[0] = -sum(C[k] for k in (1..n))
        R.append((C[0]*f).denominator())
    return R
print(A006568_list(28)) # Peter Luschny, Feb 20 2016

Given a variant of Pascal's triangle (cf. A209518) in which the two rightmost diagonals are deleted, invert the triangle and extract the leftmost column. Considered as a sequence, we obtain A006568/A006569: (1, -1/3, 1/18, 1/90, ...). - Gary W. Adamson, Mar 09 2012

More terms from Peter Luschny, Feb 20 2016

A006569 Numerators of generalized Bernoulli numbers.

Original entry on oeis.org

1, -1, 1, 1, -1, -5, -1, 7, 13, -307, -479, 1837, 100921, 15587, -23737, -5729723, 14731223, 9129833, 2722952839, -4700745901, -1556262845, 190717213397, 24684889339847, -50242799489, -148437433077277, -8592042383621, 221844330989749, 176585172615885307, -9245931549625447

Offset: 0

N. J. A. Sloane

F. T. Howard, A sequence of numbers related to the exponential function, Duke Math. J., 34 (1967), 599-615.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, Page 7, 2nd table is identical to A006569/A006568.
Abdul Hassen and Hieu D. Nguyen, Hypergeometric Zeta Functions, arXiv:math/0509637 [math.NT], Sep 27 2005.
Index entries for sequences related to Bernoulli numbers.

Cf. A006568.

Cf. A132092-A132099.

eq:=n->bernoulli(n+1)=a[n+1]-sum(binomial(n+1,r)*bernoulli(r)*a[n+2-r],r=1..n+1): a[0]:=1:for n from 0 to 28 do a[n+1]:=solve(eq(n),a[n+1]) od: seq(numer(a[n]),n=0..29); # Emeric Deutsch, Jan 23 2005

rows = 29; M = Table[If[n-1 <= k <= n, 0, Binomial[n, k]], {n, 2, rows+1}, {k, 0, rows-1}] // Inverse;
M[[All, 1]] // Numerator (* Jean-François Alcover, Jul 14 2018 *)

def A006569_list(len):
    f, R, C = 1, [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        f *= n
        for k in range(n, 0, -1):
            C[k] = C[k-1] / (k+2)
        C[0] = -sum(C[k] for k in (1..n))
        R.append((C[0]*f).numerator())
    return R
print(A006569_list(29)) # Peter Luschny, Feb 20 2016

Recurrence relation: Bernoulli(n+1) = a(n+1) - Sum_{r=1..n+1} binomial(n+1, r)*Bernoulli(r)*a(n+2-r); a(0)=1 (p. 603 of the Howard reference). - Emeric Deutsch, Jan 23 2005

E.g.f. for fractions: x^2/2 / (e^x-1-x). - Franklin T. Adams-Watters, Nov 04 2009

More terms from Emeric Deutsch, Jan 23 2005

A132093 Denominators of Blandin-Diaz compositional Bernoulli numbers (B^sin)_3,n.

Original entry on oeis.org

1, 10, 350, 1050, 57750, 250250, 2388750, 2231250, 1088106250, 137156250, 105761906250, 2289218750, 8842968750, 51289218750, 45049030468750, 3563716406250, 1099667378906250, 4714260332031250, 14142780996093750

Offset: 0

Jonathan Vos Post, Aug 09 2007

Numerators and denominators given only for even n (odd n have numerators = 0).

			-1, 0, -1/10, 0, -11/350, 0, -17,1050, 0, -563/57750, 0, -381/250250, 0.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. See Table 3.3.

Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, p. 8, arXiv:0708.0809 [math.CO], 2007-2008.

Numerators are A132092.

Cf. A132094-A132099.

m = 20;
((x^3)/3!)/(Sin[x]-x) + O[x]^(2m) // CoefficientList[#, x]& // #*Range[0, 2m-2]!& // #[[;; ;; 2]]& // Denominator (* Jean-François Alcover, Mar 23 2020 *)

my(N=40, x='x+O('x^N), v=apply(denominator, Vec(serlaplace(x^3/(6*(sin(x)-x)))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Jan 24 2024

((x^3)/3!)/(sin(x)-x) = Sum_{n>=0} (B^sin)_3,n ((x^n)/n!).

More terms from R. J. Mathar, May 25 2008

Offset corrected as suggested by Andrew Howroyd. - N. J. A. Sloane, Sep 22 2024

cp's OEIS Frontend

A001896 Numerators of cosecant numbers -2*(2^(2*n - 1) - 1)*Bernoulli(2*n); also of Bernoulli(2*n, 1/2) and Bernoulli(2*n, 1/4).

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A001897 Denominators of cosecant numbers: -2*(2^(2*n-1)-1)*Bernoulli(2*n).

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Magma

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PARI

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A132099 Denominators of Blandin-Diaz compositional Bernoulli numbers (B^Z^2)_1,n.

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A132094 Numerators of expansion of e.g.f. x^2/(2*(cos(x)-1)), even powers only.

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A132095 Denominators of expansion of e.g.f. x^2/(2*(cos(x)-1)), even powers only.

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A132097 Denominators of Blandin-Diaz compositional Bernoulli numbers (B^Z)_1,n.

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A132098 Numerators of Blandin-Diaz compositional Bernoulli numbers (B^Z^2)_1,n.

A001896 Numerators of cosecant numbers -2(2^(2n - 1) - 1)Bernoulli(2n); also of Bernoulli(2n, 1/2) and Bernoulli(2n, 1/4).

A001897 Denominators of cosecant numbers: -2(2^(2n-1)-1)Bernoulli(2n).