%I A001469 M3041 N1233
%S A001469 1,1,3,17,155,2073,38227,929569,28820619,1109652905,51943281731,
%T A001469 2905151042481,191329672483963,14655626154768697,1291885088448017715,
%U A001469 129848163681107301953,14761446733784164001387
%V A001469 -1,1,-3,17,-155,2073,-38227,929569,-28820619,1109652905,-51943281731,
%W A001469 2905151042481,-191329672483963,14655626154768697,-1291885088448017715,
%X A001469 129848163681107301953,-14761446733784164001387
%N A001469 Genocchi numbers (of first kind); unsigned coefficients give expansion
of tan(x/2).
%C A001469 The Genocchi numbers satisfy Seidel's recurrence: for n>1, 0 = sum{j=0..[n/
2], C(n,2j)*a(n-j)}. - R. Stephan, Apr 17 2004
%C A001469 The (n+1)st Genocchi number is the number of Dumont permutations of the
first kind on 2n letters. In a Dumont permutation of first kind,
each even integer must be followed by a smaller integer and each
odd integer is either followed by a larger integer or is the last
element. - R. Stephan, Apr 26 2004
%D A001469 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A001469 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A001469 R. C. Archibald, Review of Terrill-Terrill paper, Math. Comp., 1 (1945),
pp. 385-386.
%D A001469 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D A001469 D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke
Math. J., 41 (1974), 305-318.
%D A001469 Dumont, Dominique and Randrianarivony, Arthur, Sur une extension des
nombres de Genocchi, European J. Combin. 16 (1995), 147-151.
%D A001469 Dumont, Dominique and Randrianarivony, Arthur, Derangements et nombres
de Genocchi, Discrete Math. 132 (1994), 37-49.
%D A001469 R. Ehrenborg and E. Steingrimsson, Yet another triangle for the Genocchi
numbers, Europ. J. Combin., 21 (2000), 593-600.
%D A001469 L. Euler, Institutionum Calculi Differentialis, volume 2 (1755), para.
181.
%D A001469 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.
%D A001469 A. Genocchi, Intorno all'espressione generale de'numeri Bernulliani,
Ann. Sci. Mat. Fis., 3 (1852), 395-405.
%D A001469 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley,
Reading, MA, 1990, p. 528.
%D A001469 J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist,
14 (1989), 1-23.
%D A001469 G. Kreweras, Sur les permutations compte'es par les nombres de Genocchi...,
Europ. J. Comb., vol. 18, pp. 49-58, 1997.
%D A001469 D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli
and Euler, Annals Math., 36 (1935), 637-649.
%D A001469 H. M. Terrill and E. M. Terrill, Tables of numbers related to the tangent
coefficients, J. Franklin Inst., 239 (1945), 64-67.
%D A001469 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see
Problem 5.8.
%D A001469 G. Viennot, Interpretations combinatoires des nombres d'Euler et de Genocchi,
Seminar on Number Theory, 1981/1982, No. 11, 94 pp., Univ. Bordeaux
I, Talence, 1982.
%H A001469 T. D. Noe, <a href="b001469.txt">Table of n, a(n) for n=1..100</a>
%H A001469 Alexander Burstein, Sergi Elizalde and Toufik Mansour, <a href="http:/
/arXiv.org/abs/math.CO/0610234">Restricted Dumont permutations, Dyck
paths and noncrossing partitions</a>, arXiv math.CO/0610234.
%H A001469 M. Domaratzki, <a href="http://www.cs.queensu.ca/TechReports/Reports/
2001-449.ps">A Generalization of the Genocchi Numbers with Applications
to ...</a>
%H A001469 M. Domaratzki, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Combinatorial
Interpretations of a Generalization of the Genocchi Numbers</a>,
Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.6.
%H A001469 I. M. Gessel, <a href="http://www.arXiv.org/abs/math.CO/0108121">Applications
of the classical umbral calculus</a>.
%H A001469 T. Mansour, <a href="http://arXiv.org/abs/math.CO/0209379">Restricted
132-Dumont permutations</a>.
%H A001469 A. Randrianarivony and J. Zeng, <a href="http://igd.univ-lyon1.fr/home/
zeng/public_html/paper/publication.html">Une famille des polynomes
qui interpole plusieurs suites...</a>, Adv. Appl. Math. 17 (1996),
1-26.
%H A001469 H. J. H. Tuenter, <a href="http://arXiv.org/abs/math.NT/0606080">Walking
into an absolute sum</a>
%H A001469 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
GenocchiNumber.html">Link to a section of The World of Mathematics.</
a>
%H A001469 <a href="Sindx_Be.html#Bernoulli">Index entries for sequences related
to Bernoulli numbers.</a>
%F A001469 a(n) = 2*(1-4^n)*B_{2n} (B = Bernoulli numbers).
%F A001469 x tan (x/2) = Sum_{n>=1} x^(2n)*|a(n)|/(2n)! = x^2/2 + x^4/24 + x^6/240
+ 17*x^8/40320 + 31*x^10/725760 + O(x^11).
%F A001469 2x/(1 + e^x) = x + Sum_{n >= 1} a(2n) x^(2n) / (2n)! = - x^2/2! + x^4/
4! - 3 x^6/6! + 17 x^8/8! + ...
%F A001469 a(n)=sum(k=0, 2n-1, 2^k*B(k)*C(n, k)) where B(k) is the k-th Bernoulli
number and C(n, k)=binomial(n, k) - Benoit Cloitre (benoit7848c(AT)orange.fr),
May 31 2003
%F A001469 |a(n)| = Sum_{k=0..2*n} (-1)^(n-k+1)*Stirling2(2*n, k)*A059371(k). -
Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 07 2004
%p A001469 A001469 := proc(n::integer) RETURN( (2*n)!*coeftayl( 2*x/(exp(x)+1),x=0,
2*n) ) ; end: for n from 1 to 20 do print(A001469(n)) ; od : - R.
J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 22 2006
%o A001469 (PARI) a(n)=if(n<1,0,n*=2; 2*(1-2^n)*bernfrac(n))
%Y A001469 Cf. A000182, A006846. a(n)=-A065547(n, 1) and A065547(n+1, 2), n>=1.
%Y A001469 Sequence in context: A145081 A020562 A135751 this_sequence A110501 A066211
A163884
%Y A001469 Adjacent sequences: A001466 A001467 A001468 this_sequence A001470 A001471
A001472
%K A001469 sign,easy,nice
%O A001469 1,3
%A A001469 N. J. A. Sloane (njas(AT)research.att.com).
%E A001469 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 06 2000
|
