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A000806
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Bessel polynomial y_n(-1).
(Formerly M3982 N1651)
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27
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1, 0, 1, -5, 36, -329, 3655, -47844, 721315, -12310199, 234615096, -4939227215, 113836841041, -2850860253240, 77087063678521, -2238375706930349, 69466733978519340, -2294640596998068569, 80381887628910919255, -2976424482866702081004, 116160936719430292078411
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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. - Olivier Gérard, Feb 08 2011
|a(n)| is the number of perfect matchings in the complement of P_{2n} where P_{2n} is the path graph on 2n vertices. - Andrew Howroyd, Mar 15 2016
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REFERENCES
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G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université 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).
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LINKS
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Ron M. Adin, Arkady Berenstein, Jacob Greenstein, Jian-Rong Li, Avichai Marmor, and Yuval Roichman, Transitive and Gallai colorings, arXiv:2309.11203 [math.CO], 2023. See p. 6.
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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
D-finite with recurrence 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(n) = Sum_{m=0..n} A001498(n,m)*(-1)^m, n>=0 (alternating row sums of Bessel triangle).
E.g.f. for unsigned version: -exp(sqrt(1-2*x)-1). - Karol A. Penson, Mar 20 2010 [gives -1, 1, 0, 1, 5, 36, 329, ... ]
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
0 = a(n) * (a(n+2)) + a(n+1) * (-a(n+1) + 2*a(n+2) + a(n+3)) + a(n+2) * (-a(n+2)) for all n in Z. - Michael Somos, Jan 27 2014
a(n) = -i*(BesselK[3/2,1]*BesselI[n+3/2,-1] - BesselI[3/2,-1]*BesselK[n+3/2,1]), n>=0 for unsigned version - G. C. Greubel , Apr 19 2015
a(n) = hypergeom( [n+1, -n], [], 1/2). - Peter Luschny, Nov 10 2016
a(n) = (1/2)_{n} * (-2)^n * hypergeometric1f1(-n; -2*n; -2).
G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; -2*t/(1-t)^2). (End)
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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.
G.f. = 1 + x^2 - 5*x^3 + 36*x^4 - 329*x^5 + 3655*x^6 - 47844*x^7 + ...
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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
a := n -> hypergeom([n+1, -n], [], 1/2): seq(simplify(a(n)), n=0..20); # Peter Luschny, Nov 10 2016
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MATHEMATICA
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Table[Sum[Binomial[n, i]*(2*n-i)!/2^(n-i)*(-1)^(n-i)/n!, {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2013 *)
a[ n_] := With[ {m = If[ n<0, -n-1, n]}, (-1)^m (2 m - 1)!! Hypergeometric1F1[ -m, -2 m, -2] ]; (* Michael Somos, Jan 27 2014 *)
a[ n_] := With[ {m = If[ n<0, -n-1, n]}, Sum[ (-1)^(m - i) (2 m - i)! / (2^(m - i) i! (m - i)!), {i, 0, m}] ]; (* Michael Somos, Jan 27 2014 *)
a[ n_] := With[ {m = If[ n<0, -n-1, n]}, If[ m<1, 1, (-1)^m Numerator @ FromContinuedFraction[ Table[ (-1)^Quotient[k, 2] If[ OddQ[k], k, 1], {k, 2 m}] ] ] ]; (* Michael Somos, Jan 27 2014 *)
Table[(-1)^n (2 n - 1)!! Hypergeometric1F1[-n, -2 n, -2], {n, 0, 20}] (* Eric W. Weisstein, Nov 14 2018 *)
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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 */
(PARI) {a(n) = if( n<0, n=-n-1); contfracpnqn( vector( 2*n, k, (-1)^(k\2) * if( k%2, k, 1))) [1, 1] }; /* Michael Somos, Jan 27 2014 */
(Magma) I:=[0, 1]; [1] cat [n le 2 select I[n] else (1-2*n)*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Apr 19 2015
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CROSSREFS
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KEYWORD
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sign,easy,nice
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AUTHOR
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STATUS
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approved
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