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A000806
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Bessel polynomial y_n(-1).
(Formerly M3982 N1651)
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15
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1, 0, 1, -5, 36, -329, 3655, -47844, 721315, -12310199, 234615096, -4939227215, 113836841041, -2850860253240, 77087063678521, -2238375706930349, 69466733978519340, -2294640596998068569, 80381887628910919255, -2976424482866702081004
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OFFSET
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0,4
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COMMENTS
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a(n) can be seen as a subset of the unordered pairings of the first 2n integers (A001147) with forbidden pairs (i,i+1) for all i in [1,2n-1] (all adjacent integers). The circular version of this constraint is A003436. [From Olivier Gérard, Feb 8 2011]
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REFERENCES
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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.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
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).
J. Touchard, Nombres exponentiels et nombres de Bernoulli, Canad. J. Math., 8 (1956), 305-320.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
Index entries for sequences related to Bessel functions or polynomials
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FORMULA
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E.g.f.: exp(sqrt(1+2*x)-1)/sqrt(1+2*x). - Michael Somos, Feb 16, 2002
a(n) = (-2*n+1)*a(n-1) + a(n-2). - T. D. Noe, Oct 26 2006
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
a(-1-n)= a(n). - Michael Somos Apr 02 2007
a(n)=sum(A001498(n,m)*(-1)^m,m=0..n), n>=0 (alternating row sums of Bessel triangle).
E.g.f. for unsigned version: -exp(sqrt(1-2*x)-1). [From Karol A. Penson, Mar 20 2010] (gives -1, 1, 0, 1, 5, 36, 329, ... )
E.g.f. for unsigned version: 1/(sqrt(1-2*x))*exp(sqrt(1-2*x)-1). [Sergei N. Gladkovskii, Jul 03 2012]
G.f.: 1/G(0) where G(k)= 1 - x + x*(2*k+1)/(1 - x + 2*x*(k+1)/G(k+1)); (continued fraction, 2-step). [Sergei N. Gladkovskii, Jul 10 2012]
G.f.: 1+x/U(0) where U(k)= 1 - x + x*(k+1)/U(k+1) ; (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Oct 06 2012
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EXAMPLE
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For n=3, the a(3) = 5 solutions are (14) (25) (36), (14) (26) (35), (15) (24) (36), (16) (24) (35), (13) (25) (46) excluding 10 other possible pairings.
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MAPLE
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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
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MATHEMATICA
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a[n_] := a[n] = (-2n+1)*a[n-1] + a[n-2]; a[0] = 1; a[1] = 0; Table[a[n], {n, 0, 19}] (* From Jean-François Alcover, Nov 29 2011, after T. D. Noe *)
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PROG
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(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 */
(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 */
(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 */
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CROSSREFS
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Cf. A001515.
Polynomial coefficients are in A001498. Cf. A003436.
Cf. A101682.
Sequence in context: A135149 A187827 A067305 * A127132 A141764 A075744
Adjacent sequences: A000803 A000804 A000805 * A000807 A000808 A000809
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KEYWORD
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sign,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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