|
| |
|
|
A002114
|
|
Glaisher's H' numbers.
(Formerly M4810 N2057)
|
|
15
|
|
|
|
1, 11, 301, 15371, 1261501, 151846331, 25201039501, 5515342166891, 1538993024478301, 533289474412481051, 224671379367784281901, 113091403397683832932811, 67032545884354589043714301, 46211522130188693681603906171
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
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
|
|
|
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
|
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
|
|
|
FORMULA
|
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
|
|
|
MAPLE
|
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);
|
|
|
MATHEMATICA
|
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 *)
|
|
|
PROG
|
(Maxima)
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 *)
|
|
|
CROSSREFS
|
Sequence in context: A012184 A012027 A279181 * A012192 A012079 A180056
Adjacent sequences: A002111 A002112 A002113 * A002115 A002116 A002117
|
|
|
KEYWORD
|
nice,easy,nonn
|
|
|
AUTHOR
|
N. J. A. Sloane
|
|
|
STATUS
|
approved
|
| |
|
|
