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A002111 Glaisher's G numbers.
(Formerly M4007 N1660)
7
1, 5, 49, 809, 20317, 722813, 34607305, 2145998417, 167317266613, 16020403322021, 1848020950359841, 252778977216700025, 40453941942593304589, 7488583061542051450829, 1587688770629724715374457, 382218817191632327375004833 (list; graph; refs; listen; history; text; internal format)
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

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

T. D. Noe and Seiichi Manyama, Table of n, a(n) for n = 1..254 (first 50 terms from T. D. Noe)

S. Cooper, Cubic elliptic functions, Res. Lett. Inf. Math. Sci., Vol. 5, 2003, 23-59, see page 30.

J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., 31 (1899), 216-235.

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-Fran├žois 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: A062995 A104600 A221972 * A001819 A064618 A249588

Adjacent sequences:  A002108 A002109 A002110 * A002112 A002113 A002114

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 20 17:15 EDT 2017. Contains 290836 sequences.