A047788 -id:A047788 - cp's OEIS frontend

A002114 Glaisher's H' numbers.

1, 11, 301, 15371, 1261501, 151846331, 25201039501, 5515342166891, 1538993024478301, 533289474412481051, 224671379367784281901, 113091403397683832932811, 67032545884354589043714301, 46211522130188693681603906171

Offset: 1

N. J. A. Sloane

a(n) mod 9 = 1,2,4,8,7,5 repeated period 6 (A153130, see also A001370). a(n) mod 10 = 1. - Paul Curtz, Sep 10 2009

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.
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 = 1..100
J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., 31 (1899), 216-235.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010. [Author's named corrected by _N. J. A. Sloane_, Dec 12 2021]
Index entries for sequences related to Glaisher's numbers

a := n -> (-1)^n*6^(2*n)*(Zeta(0,-n*2,1/3)-Zeta(0,-n*2, 5/6)):
seq(a(n), n=1..14);

Select[Rest[With[{nn=28},CoefficientList[Series[1/(2 (2Cos[x]-1)), {x,0,nn}], x]Range[0,nn]!]],#!=0&] (* Harvey P. Dale, Jul 27 2011 *)
FullSimplify[Table[(-1)^(s+1) * BernoulliB[2*s] * (Zeta[2*s + 1, 1/6] - Zeta[2*s + 1, 5/6]) / (4*Pi*Sqrt[3]*Zeta[2*s]), {s, 1, 20}]]  (* Vaclav Kotesovec, May 05 2020 *)

a(n) := sum(sum(binomial(k,j)*(-1)^(k-j+1)*1/2^(j-1)*sum((-1)^(n)*binomial(j,i)*(2*i-j)^(2*n),i,0,floor((j-1)/2)),j,0,k)*(-2)^(k-1),k,1,2*n); /* Vladimir Kruchinin, Aug 05 2010 */

H'(n) = H(n)/3, where H(n)=2^(2n+1)*I(n) (see A002112) and e.g.f. for (-1)^n*I(n) is (3/2)/(1+exp(x)+exp(-x)) (see A047788, A047789).
H'(n) = A000436(n)/2^(2n+1). - Philippe Deléham, Jan 17 2004
For n > 0, H'(n) = Sum{k = 0..n, T(n, k)*9^(n-k)*2^(k-1) }; where DELTA is the operator defined in A084938, T(n, k) is the triangle, read by rows, given by :[0, 1, 0, 4, 0, 9, 0, 16, 0, 25, ...] DELTA [1, 0, 10, 0, 28, 0, 55, 0, 90, ..]= {1}; {0, 1}; {0, 1, 1}; {0, 1, 12, 1}; {0, 1, 63, 123, 1}; {0, 1, 274, 2366, 1234, 1}; ... For 1, 10, 28, 55, 90, 136, ... see A060544 or A060544. - Philippe Deléham, Jan 17 2004
E.g.f. 1/2*1/(2*cos(x)-1). a(n)=sum(sum(binomial(k,j)*(-1)^(k-j+1)*1/2^(j-1)*sum((-1)^(n)*binomial(j,i)*(2*i-j)^(2*n),i,0,floor((j-1)/2)),j,0,k)*(-2)^(k-1),k,1,2*n), n>0. - Vladimir Kruchinin, Aug 05 2010
E.g.f.: E(x)= x^2/(G(0)-x^2) ; G(k)= 2*(2*k+1)*(k+1) - x^2 + 2*x^2*(2*k+1)*(k+1)/G(k+1); (continued fraction Euler's kind, 1-step ). - Sergei N. Gladkovskii, Jan 03 2012
If E(x)=Sum(k=0,1,..., a(k+1)*x^(2k+2)), then A002114(k) = a(k+1)*(2*k+2)!. - Sergei N. Gladkovskii, Jan 09 2012
a(n) ~ (2*n)! * 3^(2*n+1/2) / Pi^(2*n+1). - Vaclav Kotesovec, Feb 26 2014
a(n) = (-1)^n*6^(2*n)*(zeta(-n*2,1/3)-zeta(-n*2,5/6)), where zeta(a, z) is the generalized Riemann zeta function.
From Vaclav Kotesovec, May 05 2020: (Start)
a(n) = (2*n)! * (zeta(2*n+1, 1/6) - zeta(2*n+1, 5/6)) / (sqrt(3)*(2*Pi)^(2*n+1)).
a(n) = (-1)^(n+1) * Bernoulli(2*n) * (zeta(2*n+1, 1/6) - zeta(2*n+1, 5/6)) / (4*Pi*sqrt(3)*zeta(2*n)). (End)
Conjectural e.g.f.: Sum_{n >= 1} (-1)^n*Product_{k = 1..n} (1 - exp(A007310(k)*z) ) =  z + 11*z^2/2! + 301*z^3/3! + .... - Peter Bala, Dec 09 2021

A002111 Glaisher's G numbers.

Original entry on oeis.org

1, 5, 49, 809, 20317, 722813, 34607305, 2145998417, 167317266613, 16020403322021, 1848020950359841, 252778977216700025, 40453941942593304589, 7488583061542051450829, 1587688770629724715374457, 382218817191632327375004833

Offset: 1

N. J. A. Sloane

Related to the formula Sum_{k>0} sin(kx)/k^(2n+1)=(-1)^(n+1)/2*x^(2n+1)/(2n+1)! * Sum_{i=0..2n} (2Pi/x)^i*B(i)*C(2n+1,i). - Benoit Cloitre, May 01 2002

Named after the English mathematician and astronomer James Whitbread Lee Glaisher (1848-1928). - Amiram Eldar, Jun 16 2021

			G.f. = x + 5*x^2 + 49*x^3 + 809*x^4 + 20317*x^5 + 722813*x^6 + 34607305*x^7 + ...

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.
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).

Seiichi Manyama, Table of n, a(n) for n = 1..254 (first 50 terms from T. D. Noe)
Shaun Cooper, Cubic elliptic functions, Res. Lett. Inf. Math. Sci., Vol. 5 (2003), pp. 23-59, see page 30.
Ira M. Gessel, On the Almkvist-Meurman Theorem for Bernoulli Polynomials, Integers (2023) Vol. 23, #A14.
J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., Vol. 31 (1899), pp. 216-235.
René Gy, Bernoulli-Stirling Numbers, Integers, Vol. 20, (2020), #A67.
J. C. P. Miller, Letter to N. J. A. Sloane, Mar 26 1971.
N. J. A. Sloane, Transforms.
Wikipedia, Bernoulli Polynomials.
Index entries for sequences related to Glaisher's numbers

Cf. A083007, A033470.

read transforms; t1 := (3/2)/(1+exp(x)+exp(-x)); series(t1,x,50): t2 := SERIESTOLISTMULT(t1); [seq(n*t2[n],n=1..nops(t5))];

s[n_] := CoefficientList[Series[(1/2)*(Sin[t/2]/Sin[3*(t/2)]), {t, 0, 32}], t][[n + 1]]*n!*(-1)^Floor[n/2]; a[n_] := (-1)^n*(6*n + 3)*s[2*n]; Table[a[n], {n, 1, 16}] (* Jean-François Alcover, Mar 22 2011, after Michael Somos' formula *)
a[ n_] := If[ n < 1, 0, (2 n + 1)! SeriesCoefficient[ 3 / (2 + 4 Cos[x]), {x, 0, 2 n}]]; (* Michael Somos, Jun 01 2012 *)

{a(n) = if( n<1, 0, n*=2; (n+1)! * polcoeff( 3 / (2 + 4 * cos( x + O(x^n))), n))}; /* Michael Somos, Feb 26 2004 */

a(n)=if(n<1,0,-(-1)^n*sum(i=0,2*n,binomial(2*n+1,i)*bernfrac(i)*3^i)) \\ Benoit Cloitre, May 01 2002

def A002111(n):
    return add(add(add(((-1)^(n+1-v)/(j+1))*binomial(2*n+1,k)*binomial(j,v)*(3*v)^k for v in (0..j)) for j in (0..k)) for k in (0..2*n+1))
[A002111(n) for n in (1..16)]  # Peter Luschny, Jun 03 2013

To get these numbers, expand the e.g.f. (3/2)/(1+exp(x)+exp(-x)), multiply coefficient of x^n by (n+1)! and take absolute values.
Or expand the e.g.f. (3/2)/(1+2*cos(x)) and multiply coefficient of x^n by (n+1)!. - Herb Conn, Feb 25 2002
a(n) = (2n+1)*I(n), where I(n) is given by A047788/A047789.
a(n) = Sum_{i=0, 2n} B(i)*C(2n+1, i)*3^i where B(i) are the Bernoulli numbers, C(2n, i) the binomial numbers. - Benoit Cloitre, May 01 2002
a(n) = (-1)^n * (6*n + 3) * s(2*n), if n>0, where s(n) are the cubic Bernoulli numbers. - Michael Somos, Feb 26 2004
E.g.f.: 3*x / (2 + 4*cos(x)) = Sum_{n>=0} a(n) * x^(2*n+1) / (2*n+1)!. - Michael Somos, Feb 26 2004
E.g.f.: E(x) = (3/2)/(1+2*cos(x)) - 1/2 = x^2/(3*G(0)+x^2); G(k) = 2*(2*k+1)*(k+1) - x^2 + 2*x^2*(2*k+1)*(k+1)/G(k+1); (continued fraction Euler's kind, 1-step). Let f[n]:=coeftayl(E(x), x=0, n) then: A002111[n]=f[2*n+2]*((2*n+3)!). - Sergei N. Gladkovskii, Jan 14 2012
a(n) = Sum_{k=0..2n+1} Sum_{j=0..k} Sum_{v=0..j} ((-1)^(n-v+1)/(j+1))* binomial(2*n+1,k)*binomial(j,v)*(3*v)^k. - Peter Luschny, Jun 03 2013
a(n) ~ (2*n+1)! * sqrt(3) * (3/(2*Pi))^(2*n+1). - Vaclav Kotesovec, Jul 30 2013
From Peter Bala, Mar 02 2015: (Start)
a(n) = (-1)^(n+1)*3^(2*n+1)*B(2*n+1,1/3), where B(n,x) denotes the n-th Bernoulli polynomial. Cf. A009843, A069852, A069994.
Conjecturally, a(n) = the unsigned numerator of B(2*n+1,1/3). Cf. A033470.
Essentially a bisection of |A083007|.
G.f. for signed version of sequence: 1/2 + 1/2*Sum_{n >= 0} { 1/(n+1) * Sum_{k = 0..n} (-1)^(k+1)*binomial(n,k)/( (1 - (3*k + 1)*x)*(1 - (3*k + 2)*x) ) } = x^2 - 5*x^4 + 49*x^6 - .... (End)

A047789 Denominators of Glaisher's I-numbers.

Original entry on oeis.org

2, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 81, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 1, 1, 3, 1, 1, 3, 1

Offset: 0

N. J. A. Sloane

			1/2, 1/3, 1, 7, 809/9, 1847, 55601, 6921461/3,...

Robert Israel, Table of n, a(n) for n = 0..10000
J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., 31 (1899), 216-235.
Index entries for sequences related to Glaisher's numbers

Cf. A047788, A002111.

f:= n -> 3^padic:-ordp(2*n+1,3):
f(0):= 2:
map(f, [$0..200]); # Robert Israel, Aug 14 2018

a[0] = 2; a[n_] := 3^IntegerExponent[2n+1, 3];
Table[a[n], {n, 0, 101}] (* Jean-François Alcover, Feb 27 2019 *)
a[0]:=2; a[n_]:=Denominator[FunctionExpand[(PolyGamma[2*n, 1/3] + (3^(2*n+1)-1)*(2*n)!*Zeta[2*n+1]/2)*Sqrt[3]/(-2^(2*n)*Pi^(2*n+1))]]; Table[a[n], {n,0,100}] (* Detlef Meya, Sep 28 2024 *)

a(n)=if(n<1,2*(n==0),3^valuation(2*n+1,3)) /* Michael Somos, Feb 26 2004 */

a(n)=if(n<1,2*(n==0),n*=2;denominator(n!*polcoeff(3/(2+4*cos(x+O(x^n))),n))) /* Michael Somos, Feb 26 2004 */

From Robert Israel, Aug 14 2018: (Start)
For n >= 1, a(3*n) = a(3*n+2) = 1 and a(3*n+1) = 3*a(n).
G.f. g(x) satisfies g(x) = 3*x*g(x^3) + 2 - 3*x + (x^2+x^3)/(1-x^3). (End)
G.f.: 1 + Sum_{k>=0} (3^k*x^((5*3^k - 1)/2) + 3^k*x^((3^k - 1)/2))/(1 - x^(3^(k + 1))). - Miles Wilson, Dec 01 2024

A002112 Glaisher's H numbers.

Original entry on oeis.org

3, 33, 903, 46113, 3784503, 455538993, 75603118503, 16546026500673, 4616979073434903, 1599868423237443153, 674014138103352845703, 339274210193051498798433, 201097637653063767131142903, 138634566390566081044811718513

Offset: 1

N. J. A. Sloane

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.
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 = 1..100
J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., 31 (1899), 216-235.
Michael E. Hoffman, Derivative polynomials, Euler polynomials, and associated integer sequences, The Electronic Journal of Combinatorics 6.1 (1999).
Index entries for sequences related to Glaisher's numbers

e[0] = 1; e[n_] := e[n] = (-1)^n*(1 - Sum[(-1)^i*Binomial[2n, 2i]*3^(2n-2i)*e[i], {i, 0, n-1}]); a[n_] := 3*e[n]/2^(2n+1); Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Jan 31 2012, after Philippe Deléham *)

H(n) = 2^(2n+1)*I(n), where e.g.f. for (-1)^n*I(n) is (3/2)/(1+exp(x)+exp(-x)) (see A047788, A047789).
H(n) = 3*A000436(n)/2^(2n+1) = 3*A002114(n). - Philippe Deléham, Jan 17 2004
E.g.f.: E(x) = 3*x^2/(G(0)-x^2); G(k) = 2*(2*k+1)*(k+1) - x^2 + 2*x^2*(2*k+1)*(k+1)/G(k+1); (continued fraction Euler's kind, 1-step ). - Sergei N. Gladkovskii, Jan 03 2012
If E(x) = Sum_{k>=0} a(k+1)*x^(2k+2), then A002112(k) = a(k+1)*(2*k+2)!. - Sergei N. Gladkovskii, Jan 09 2012
From Vaclav Kotesovec, May 05 2020: (Start)
a(n) = sqrt(3) * (2*n)! * (zeta(2*n+1, 1/6) - zeta(2*n+1, 5/6)) / (2*Pi)^(2*n+1).
a(n) = (-1)^(n+1) * sqrt(3) * Bernoulli(2*n) * (zeta(2*n+1, 1/6) - zeta(2*n+1, 5/6)) / (4*Pi*zeta(2*n)). (End)

A239902 Exceptional primes: those for which Eq. (4.8) in Cosgrave and Dilcher (2011) fails.

Original entry on oeis.org

13, 181, 2521, 76543, 489061, 6811741, 1321442641, 18405321661, 381765135195632792959100810331957408101589361

Offset: 1

N. J. A. Sloane, Apr 06 2014

Comments from Christopher M. Stokes, Aug 02 2022: (Start)
Also the primes p for which A047788(p-1) = 0 mod p^2.
Also the primes for which the cyclotomic lambda invariant of Q(sqrt{-3}) is greater than 1. (End)

J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.6.

J. B. Cosgrave and K. Dilcher, The multiplicative orders of certain Gauss factorials, Intl. J. Number Theory 7 (1) (2011) 145-171.
John B. Cosgrave and Karl Dilcher, The multiplicative orders of certain Gauss factorials II, Funct. Approx. Comment. Math. Volume 54, Number 1 (2016), 73-93.
D. S. Dummit, D. Ford, H. Kisilevsky, and J. W. Sands, Computation of Iwasawa lambda invariants for imaginary quadratic fields, Journal of number theory, 37(1) (1991), 100-121. [Reference added by _N. J. A. Sloane_, Jun 24 2022]
Christopher Stokes, On Gauss factorials and their application to Iwasawa theory for imaginary quadratic fields, arXiv:2207.07804 [math.NT], 2022.

Cf. A354772, A117808.

More terms from Cosgrave (2022), Section 18.6 added by N. J. A. Sloane, May 29 2022

a(9) from Stokes (2022) added by Michel Marcus, Jul 20 2022

cp's OEIS Frontend

A002114 Glaisher's H' numbers.

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A002111 Glaisher's G numbers.

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A047789 Denominators of Glaisher's I-numbers.

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A002112 Glaisher's H numbers.

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A239902 Exceptional primes: those for which Eq. (4.8) in Cosgrave and Dilcher (2011) fails.

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