A002111 Glaisher's G numbers. (Formerly M4007 N1660)

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

Offset

1,2

Comments

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

References

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).

Links

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

Formula

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)

Example

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

Maple

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))];

Mathematica

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 *)

Prog

(PARI) {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 */
(PARI) 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
(Sage)
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

Crossrefs

Cf. A083007, A033470.

Sequence in context: A293847 A104600 A221972 * A305114 A001819 A064618

Adjacent sequences: A002108 A002109 A002110 * A002112 A002113 A002114

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved