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Search: id:A002111
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| A002111 |
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Glaisher's G numbers.
(Formerly M4007 N1660)
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+0
4
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| 1, 5, 49, 809, 20317, 722813, 34607305, 2145998417, 167317266613, 16020403322021, 1848020950359841, 252778977216700025, 40453941942593304589, 7488583061542051450829, 1587688770629724715374457, 382218817191632327375004833
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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 (benoit7848c(AT)orange.fr), May 01 2002
a(n)=(-1)^n(6n+3)s(2n), if n>0, where s(n) are the cubic Bernoulli numbers. - Michael Somos Feb 26 2004
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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.
J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., 31 (1899), 216-235.
S. Cooper, Cubic elliptic functions, Res. Lett. Inf. Math. Sci., Vol. 5, 2003, 23-59, see page 30.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..50
N. J. A. Sloane, Transforms
Index entries for sequences related to Glaisher's numbers
S. Cooper, Cubic elliptic functions.
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FORMULA
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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
(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 (benoit7848c(AT)orange.fr), May 01 2002
E.g.f.: 3x/(2+4cos(x))-x/2 = Sum_{n>=0} a(n)x^(2n+1)/(2n+1)!. - Michael Somos Feb 26 2004
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MAPLE
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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))];
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PROGRAM
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(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 (benoit7848c(AT)orange.fr), May 01 2002
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CROSSREFS
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Sequence in context: A145088 A062995 A104600 this_sequence A001819 A064618 A075986
Adjacent sequences: A002108 A002109 A002110 this_sequence A002112 A002113 A002114
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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