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A002425
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Denominator of Pi^(2n)/(GAMMA(2n)*(1-2^(-2n))*Zeta(2n)).
(Formerly M5036 N2174)
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20
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1, 1, 1, 17, 31, 691, 5461, 929569, 3202291, 221930581, 4722116521, 968383680827, 14717667114151, 2093660879252671, 86125672563201181, 129848163681107301953, 868320396104950823611, 209390615747646519456961
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Consider the C(k)-summation process for divergent series: the series Sum((-1)^n*(n+1)^k)==1-2^k+3^k-4^k+..., summable C(1) to the value 1/2 for k=0, is for each k>=1 exactly summable C(k+1) to the sum s(k+1)=(2^(k+1)-1)*B(k+1)/(k+1) and so a(n)=Abs(numerator(s(2n))). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 27 2002
Odd part of tangent numbers A000182 (even part is 2^A101921(n)). - Ralf Stephan, Dec 21 2004
(-1)^n*a(n) is the numerator of Euler(2n+1,1). - N. J. A. Sloane, Nov 10 2009
a(n) is the absolute value of the constant term of the Euler polynomial E_{2n-1} times the even part of 2n. - Peter Luschny, Nov 26 2010
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REFERENCES
| H. Cohn, Bull. Am. Math. Soc., Sept. 1965, 681ff, esp. p. 688.
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. 73.
S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. Math. 46 (1914), 33-51.
Konrad Knopp, Theory and application of infinite series, Divergent series, Dover, p. 479
L. Oettinger, Archiv. Math. Phys., 26 (1856), see esp. p. 5.
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).
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 1..300
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
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FORMULA
| a(n)=(-1)^n/n*(1-4^n)*B(2*n)*2^A001511(n) where B(k) denotes the k-th Bernoulli number - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 30 2003
This is different from the sequence of numerators of the expansion of cosec(x)-cot(x) - see A089171.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009: (Start)
a(n) = denominator(4*n/((2^(2*n)-1)*bernoulli(2*n)))
(End)
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MAPLE
| A002425 := n -> (-1)^n*euler(2*n-1, 0)*2^padic[ordp](2*n, 2); [Peter Luschny, Nov 26 2010]
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MATHEMATICA
| a[n_] := (-1)^n * Numerator[ EulerE[2n + 1, 1]]; Table[ a[n], {n, 0, 17}] (* From Jean-François Alcover, Sep 20 2011, after N. J. A. Sloane's comment *)
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PROG
| (PARI) for(n=1, 20, print1(abs(numerator(2*bernfrac(2*n)*(4^n-1)/(2*n))), ", "))
(PARI) a(n)=if(n<1, 0, (-1)^n/n*(1-4^n)*bernfrac(2*n)*2^valuation(2*n, 2))
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CROSSREFS
| Numerator given by A037239.
Different from A089171.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009: (Start)
Equals A160469(n)/A048896(n-1)
Equals A089171(n)*A089170(n-1)
(End)
Sequence in context: A146667 A146462 A089171 * A046990 A059212 A058899
Adjacent sequences: A002422 A002423 A002424 * A002426 A002427 A002428
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KEYWORD
| nonn,frac,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| The n=15 term was formerly incorrectly given as 86125672563301143.
Formula and cross-references edited by Johannes W. Meijer (meijgia(AT)hotmail.com), May 21 2009
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