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Glaisher's G numbers.
(Formerly M4007 N1660)
10

%I M4007 N1660 #96 Oct 28 2023 11:24:37

%S 1,5,49,809,20317,722813,34607305,2145998417,167317266613,

%T 16020403322021,1848020950359841,252778977216700025,

%U 40453941942593304589,7488583061542051450829,1587688770629724715374457,382218817191632327375004833

%N Glaisher's G numbers.

%C 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

%C Named after the English mathematician and astronomer James Whitbread Lee Glaisher (1848-1928). - _Amiram Eldar_, Jun 16 2021

%D 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.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H Seiichi Manyama, <a href="/A002111/b002111.txt">Table of n, a(n) for n = 1..254</a> (first 50 terms from T. D. Noe)

%H Shaun Cooper, <a href="http://hdl.handle.net/10179/4407">Cubic elliptic functions</a>, Res. Lett. Inf. Math. Sci., Vol. 5 (2003), pp. 23-59, see page 30.

%H Ira M. Gessel, <a href="https://doi.org/10.5281/zenodo.7625111">On the Almkvist-Meurman Theorem for Bernoulli Polynomials</a>, Integers (2023) Vol. 23, #A14.

%H J. W. L. Glaisher, <a href="https://doi.org/10.1112/plms/s1-31.1.216">On a set of coefficients analogous to the Eulerian numbers</a>, Proc. London Math. Soc., Vol. 31 (1899), pp. 216-235.

%H René Gy, <a href="http://math.colgate.edu/~integers/u67/u67.pdf">Bernoulli-Stirling Numbers</a>, Integers, Vol. 20, (2020), #A67.

%H J. C. P. Miller, <a href="/A002439/a002439_1.pdf">Letter to N. J. A. Sloane, Mar 26 1971</a>.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_polynomials">Bernoulli Polynomials</a>.

%H <a href="/index/Ge#Glaisher">Index entries for sequences related to Glaisher's numbers</a>

%F 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.

%F 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

%F a(n) = (2n+1)*I(n), where I(n) is given by A047788/A047789.

%F 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

%F 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

%F 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

%F 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

%F 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

%F a(n) ~ (2*n+1)! * sqrt(3) * (3/(2*Pi))^(2*n+1). - _Vaclav Kotesovec_, Jul 30 2013

%F From _Peter Bala_, Mar 02 2015: (Start)

%F 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.

%F Conjecturally, a(n) = the unsigned numerator of B(2*n+1,1/3). Cf. A033470.

%F Essentially a bisection of |A083007|.

%F 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)

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

%p 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))];

%t 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 *)

%t a[ n_] := If[ n < 1, 0, (2 n + 1)! SeriesCoefficient[ 3 / (2 + 4 Cos[x]), {x, 0, 2 n}]]; (* _Michael Somos_, Jun 01 2012 *)

%o (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 */

%o (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

%o (Sage)

%o def A002111(n):

%o 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))

%o [A002111(n) for n in (1..16)] # _Peter Luschny_, Jun 03 2013

%Y Cf. A083007, A033470.

%K nonn,nice,easy

%O 1,2

%A _N. J. A. Sloane_