Search: id:A000806
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%I A000806 M3982 N1651
%S A000806 1,0,1,5,36,329,3655,47844,721315,12310199,234615096,4939227215,
%T A000806 113836841041,2850860253240,77087063678521,2238375706930349,69466733978519340,
%U A000806 2294640596998068569,80381887628910919255,2976424482866702081004
%V A000806 1,0,1,-5,36,-329,3655,-47844,721315,-12310199,234615096,-4939227215,
%W A000806 113836841041,-2850860253240,77087063678521,-2238375706930349,69466733978519340,
%X A000806 -2294640596998068569,80381887628910919255,-2976424482866702081004
%N A000806 Bessel polynomial y_n(-1).
%D A000806 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000806 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000806 G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble 
               fini totalement ordonne, Publications de l'Institut de Statistique 
               de l'Universit\'{e} de Paris, 23 (1978), 57-74.
%D A000806 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%D A000806 J. Touchard, Nombres exponentiels et nombres de Bernoulli, Canad. J. 
               Math., 8 (1956), 305-320.
%H A000806 T. D. Noe, <a href="b000806.txt">Table of n, a(n) for n=0..100</a>
%H A000806 <a href="Sindx_Be.html#Bessel">Index entries for sequences related to 
               Bessel functions or polynomials</a>
%F A000806 E.g.f.: exp(sqrt(1+2*x)-1)/sqrt(1+2*x). - Michael Somos, Feb 16, 2002
%F A000806 a(n) = (-2*n+1)*a(n-1) + a(n-2). - T. D. Noe, Oct 26 2006
%F A000806 If y = x + Sum_{k>1} A000272(k)*x^k/k!, then y = x + Sum{k>1} a(k-2)(-y)^k/
               k!. - Michael Somos Sep 07 2005
%F A000806 a(-1-n)= a(n). - Michael Somos Apr 02 2007
%F A000806 a(n)=sum(A001498(n,m)*(-1)^m,m=0..n), n>=0 (alternating row sums of Bessel 
               triangle).
%p A000806 A000806 := proc(n) option remember; if n<=1 then n else (2*n+1)*A000806(n-1)+A000806(n-2); 
               fi; end; # for unsigned version
%o A000806 (PARI) {a(n)= if(n<0, n= -n-1); sum(k=0, n, (2*n-k)!/(k!* (n-k)!)* (-1/
               2)^(n-k) )} /* Michael Somos Apr 02 2007 */
%o A000806 (PARI) {a(n)= local(A); if(n<0, n= -n-1); A= sqrt(1 +2*x +x*O(x^n)); 
               n!*polcoeff( exp(A-1)/A, n)} /* Michael Somos Apr 02 2007 */
%o A000806 (PARI) {a(n)= local(A); if(n<0, n= -n-1); n+=2; -(-1)^n*n!* polcoeff( 
               serreverse( sum(k=1, n, k^(k-2)* x^k/k!, x*O(x^n))), n)} /* Michael 
               Somos Apr 02 2007 */
%Y A000806 Cf. A001515.
%Y A000806 Polynomial coefficients are in A001498. Cf. A003436.
%Y A000806 Cf. A101682.
%Y A000806 Sequence in context: A091161 A135149 A067305 this_sequence A127132 A141764 
               A075744
%Y A000806 Adjacent sequences: A000803 A000804 A000805 this_sequence A000807 A000808 
               A000809
%K A000806 sign,easy,nice
%O A000806 0,4
%A A000806 N. J. A. Sloane (njas(AT)research.att.com).

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