A002111 Glaisher's G numbers.
1, 5, 49, 809, 20317, 722813, 34607305, 2145998417, 167317266613, 16020403322021, 1848020950359841, 252778977216700025, 40453941942593304589, 7488583061542051450829, 1587688770629724715374457, 382218817191632327375004833
Offset: 1
Examples
G.f. = x + 5*x^2 + 49*x^3 + 809*x^4 + 20317*x^5 + 722813*x^6 + 34607305*x^7 + ...
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
Programs
-
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 *) -
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
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) = 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)
Comments