Search: id:A110501
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%I A110501
%S A110501 1,1,3,17,155,2073,38227,929569,28820619,1109652905,51943281731,
%T A110501 2905151042481,191329672483963,14655626154768697,1291885088448017715,
%U A110501 129848163681107301953,14761446733784164001387
%N A110501 Unsigned Genocchi numbers (of first kind) of even index.
%C A110501 The Genocchi numbers satisfy Seidel's recurrence: for n>1, 0 = sum{j=0..[n/
               2], (-1)^j*C(n,2j)*a(n-j)}. - R. Stephan, Apr 17 2004
%C A110501 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 A110501 R. C. Archibald, Review of Terrill-Terrill paper, Math. Comp., 1 (1945), 
               pp. 385-386.
%D A110501 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D A110501 D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke 
               Math. J., 41 (1974), 305-318.
%D A110501 L. Euler, Institutionum Calculi Differentialis, volume 2 (1755), para. 
               181.
%D A110501 A. Genocchi, Intorno all'espressione generale de'numeri Bernulliani, 
               Ann. Sci. Mat. Fis., 3 (1852), 395-405.
%D A110501 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; p. 
               74 see Problem 5.8.
%D A110501 H. M. Terrill and E. M. Terrill, Tables of numbers related to the tangent 
               coefficients, J. Franklin Inst., 239 (1945), 64-67.
%F A110501 a(n) = 2*(-1)^n(1-4^n)*B_{2n} (B = Bernoulli numbers).
%F A110501 A002105(n) = 2^(n-1)/n * a(n). - D. E. Knuth, Jan 16 2007
%F A110501 E.g.f.: x tan(x/2) = Sum_{k > 0} a(k) x^(2k)/(2k)!.
%F A110501 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
%F A110501 O.g.f.: A(x) = x/(1-x/(1-2*x/(1-4*x/(1-6*x/(1-9*x/(1-12*x/(... -[(n+1)/
               2]*[(n+2)/2]*x/(1- ...)))))))) (continued fraction). - Paul D. Hanna 
               (pauldhanna(AT)juno.com), Jan 16 2006
%o A110501 (PARI) a(n)=if(n<1, 0, 2*(-1)^n*(1-4^n)*bernfrac(2*n))
%o A110501 (PARI) {a(n)=if(n<1, 0, (2*n)!*polcoeff( x*tan(x/2+x*O(x^(2*n))), 2*n))}
%Y A110501 Cf. A036968(2n)=A001469(n)=(-1)^n a(n).
%Y A110501 Sequence in context: A020562 A135751 A001469 this_sequence A066211 A163884 
               A052143
%Y A110501 Adjacent sequences: A110498 A110499 A110500 this_sequence A110502 A110503 
               A110504
%K A110501 nonn
%O A110501 1,3
%A A110501 Michael Somos, Jul 23 2005

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