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A000806 Bessel polynomial y_n(-1).
(Formerly M3982 N1651)
25
1, 0, 1, -5, 36, -329, 3655, -47844, 721315, -12310199, 234615096, -4939227215, 113836841041, -2850860253240, 77087063678521, -2238375706930349, 69466733978519340, -2294640596998068569, 80381887628910919255, -2976424482866702081004, 116160936719430292078411 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

The unsigned version of these numbers now has its own entry: see A278990. - N. J. A. Sloane, Dec 07 2016

REFERENCES

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).

LINKS

T. D. Noe and Seiichi Manyama, Table of n, a(n) for n = 0..404 (first 101 terms from T. D. Noe)

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. (Annotated scanned copy)

R. J. Mathar, A class of multinomial permutations avoiding object clusters, vixra:1511.0015 (2015), sequence M_{c,2}/c!.

J. Riordan, Letter to N. J. A. Sloane, Aug. 1970

Everett Sullivan, Linear chord diagrams with long chords, arXiv preprint arXiv:1611.02771 [math.CO], 2016.

J. Touchard, Nombres exponentiels et nombres de Bernoulli, Canad. J. Math., 8 (1956), 305-320.

Index entries for sequences related to Bessel functions or polynomials

FORMULA

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_{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, ... ]

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

a(n) = BesselK[n+1/2,-1]/BesselK[5/2,-1]. - Vaclav Kotesovec, Aug 07 2013

|a(n)| ~ 2^(n+1/2)*n^n/exp(n+1). - Vaclav Kotesovec, Aug 07 2013

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

From G. C. Greubel, Aug 16 2017: (Start)

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)

EXAMPLE

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 + ...

MAPLE

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

MATHEMATICA

a[n_] := a[n] = (-2n+1)*a[n-1] + a[n-2]; a[0] = 1; a[1] = 0; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 29 2011, after T. D. Noe *)

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 *)

PROG

(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

CROSSREFS

Polynomial coefficients are in A001498. Cf. A003436.

Cf. A001515, A101682, A278990.

Sequence in context: A291688 A300987 A067305 * A278990 A127132 A141764

Adjacent sequences:  A000803 A000804 A000805 * A000807 A000808 A000809

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 26 08:14 EDT 2019. Contains 322472 sequences. (Running on oeis4.)