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%I M4444 N1880 #99 Feb 18 2023 14:26:36
%S 1,-1,7,-71,1001,-18089,398959,-10391023,312129649,-10622799089,
%T 403978495031,-16977719590391,781379079653017,-39085931702241241,
%U 2111421691000680031,-122501544009741683039,7597207150294985028449,-501538173463478753560673
%N Bessel polynomial y_n(-2).
%C Absolute values give denominators of successive convergents to e using continued fraction 1+2/(1+1/(6+1/(10+1/(14+1/(18+1/(22+1/26...)))))).
%C Absolute values give number of different arrangements of nonnegative integers on a set of n 6-sided dice such that the dice can add to any integer from 0 to 6^n-1. For example when n=2, there are 7 arrangements that can result in any total from 0 to 35. Cf. A273013. The number of sides on the dice only needs to be the product of two distinct primes, of which 6 is the first example. - _Elliott Line_, Jun 10 2016
%C Absolute values give number of Krasner factorizations of (x^(6^n)-1)/(x-1) into n polynomials p_i(x), i=1,2,...,n, satisfying p_i(1)=6. In these expressions 6 can be replaced with any product of two distinct primes (Krasner and Ranulac, 1937). - _William P. Orrick_, Jan 18 2023
%C Absolute values give number of pairs (s, b) where s is a covering of the 1 X 2n grid with 1 X 2 dimers and equal numbers of red and blue 1 X 1 monomers and b is a bijection between the red monomers and the blue monomers that does not map adjacent monomers to each other. Ilya Gutkovskiy's formula counts such pairs by an inclusion-exclusion argument. The correspondence with Elliott Line's dice problem is that a dimer corresponds to a die containing an arithmetic progression of length 6 and a pair (r, b(r)), where r is a red monomer and b(r) its image under b, corresponds to a die containing the sum of an arithmetic progression of length 2 and an arithmetic progression of length 3. - _William P. Orrick_, Jan 19 2023
%D L. Euler, 1737.
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002119/b002119.txt">Table of n, a(n) for n = 0..100</a>
%H Leo Chao, Paul DesJarlais and John L Leonard, <a href="http://www.jstor.org/stable/3621238">A binomial identity, via derangements</a>, Math. Gaz. 89 (2005), 268-270.
%H J. W. L. Glaisher, <a href="https://archive.org/stream/reportofbritisha72brit#page/n345/mode/2up">On Lambert's proof of the irrationality of Pi and on the irrationality of certain other quantities</a>, Reports of British Assoc. Adv. Sci., 1871, pp. 16-18.
%H M. Krasner and B. Ranulac, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k31562/f397.item">Sur une propriété des polynomes de la division du cercle</a>, Comptes Rendus Académie des Sciences Paris, 240:397-399, 1937.
%H D. H. Lehmer, <a href="http://www.jstor.org/stable/1968107">Arithmetical periodicities of Bessel functions</a>, Annals of Mathematics, 33 (1932): 143-150. The sequence but with all signs positive is on page 149.
%H D. H. Lehmer, <a href="http://dx.doi.org/10.1090/S0025-5718-46-99625-1">Review of various tables by P. Pederson</a>, Math. Comp., 2 (1946), 68-69.
%H <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%F D-finite with recurrence a(n) = -2(2n-1)*a(n-1) + a(n-2). - _T. D. Noe_, Oct 26 2006
%F If y = x + Sum_{k>=2} A005363(k)*x^k/k!, then y = x + Sum_{k>=2} a(k-2)(-y)^k/k!. - _Michael Somos_, Apr 02 2007
%F a(-n-1) = a(n). - _Michael Somos_, Apr 02 2007
%F a(n) = (1/n!)*Integral_{x>=-1} (-x*(1+x))^n*exp(-(1+x)). - _Paul Barry_, Apr 19 2010
%F G.f.: 1/Q(0), where Q(k) = 1 - x + 2*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 17 2013
%F Expansion of exp(x) in powers of y = x*(1 + x): exp(x) = 1 + y - y^2/2! + 7*y^3/3! - 71*y^4/4! + 1001*y^5/5! - .... E.g.f.: (1/sqrt(4*x + 1))*exp(sqrt(4*x + 1)/2 - 1/2) = 1 - x + 7*x^2/2! - 71*x^3/3! + .... - _Peter Bala_, Dec 15 2013
%F a(n) = hypergeom([-n, n+1], [], 1). - _Peter Luschny_, Oct 17 2014
%F a(n) = sqrt(Pi/exp(1)) * BesselI(1/2+n, 1/2) + (-1)^n * BesselK(1/2+n, 1/2) / sqrt(exp(1)*Pi). - _Vaclav Kotesovec_, Jul 22 2015
%F a(n) ~ (-1)^n * 2^(2*n+1/2) * n^n / exp(n+1/2). - _Vaclav Kotesovec_, Jul 22 2015
%F From _G. C. Greubel_, Aug 16 2017: (Start)
%F G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; -4*t/(1-t)^2).
%F E.g.f.: (1+4*t)^(-1/2) * exp((sqrt(1+4*t) - 1)/2). (End)
%F a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*binomial(n+k,k)*k!. - _Ilya Gutkovskiy_, Nov 24 2017
%F a(n) = (-1)^n*KummerU(-n, -2*n, -1). - _Peter Luschny_, May 10 2022
%e Example from _William P. Orrick_, Jan 19 2023: (Start)
%e For n=2 the Bessel polynomial is y_2(x) = 1 + 3x + 3x^2 which satisfies y_2(-2) = -7.
%e The |a(2)|=7 dice pairs are
%e {{0,1,2,3,4,5}, {0,6,12,18,24,30}},
%e {{0,1,2,18,19,20}, {0,3,6,9,12,15}},
%e {{0,1,2,9,10,11}, {0,3,6,18,21,24}},
%e {{0,1,6,7,12,13}, {0,2,4,18,20,22}},
%e {{0,1,12,13,24,25}, {0,2,4,6,8,10}},
%e {{0,1,2,6,7,8}, {0,3,12,15,24,27}},
%e {{0,1,4,5,8,9}, {0,2,12,14,24,26}}.
%e The corresponding Krasner factorizations of (x^36-1)/(x-1) are
%e {(x^6-1)/(x-1), (x^36-1)/(x^6-1)},
%e {((x^36-1)/(x^18-1))*((x^3-1)/(x-1)), (x^18-1)/(x^3-1)},
%e {((x^18-1)/(x^9-1))*((x^3-1)/(x-1)), ((x^36-1)/(x^18-1))*((x^9-1)/(x^3-1))},
%e {((x^18-1)/(x^6-1))*((x^2-1)/(x-1)), ((x^36-1)/(x^18-1))*((x^6-1)/(x^2-1))},
%e {((x^36-1)/(x^12-1))*((x^2-1)/(x-1)), (x^12-1)/(x^2-1)},
%e {((x^12-1)/(x^6-1))*((x^3-1)/(x-1)), ((x^36-1)/(x^12-1))*((x^6-1)/(x^3-1))},
%e {((x^12-1)/(x^4-1))*((x^2-1)/(x-1)), ((x^36-1)/(x^12-1))*((x^4-1)/(x^2-1))}.
%e The corresponding monomer-dimer configurations, with dimers, red monomers, and blue monomers represented by the symbols '=', 'R', and 'B', and bijections between red and blue monomers given as sets of ordered pairs, are
%e (==, {}),
%e (B=R, {(3,1)}),
%e (BBRR, {(3,1),(4,2)}),
%e (RBBR, {((1,3),(4,2)}),
%e (R=B, {(1,3)}),
%e (BRRB, {(2,4),(3,1)}),
%e (RRBB, {(1,3),(2,4)}).
%e (End)
%p f:=proc(n) option remember; if n <= 1 then 1 else f(n-2)+(4*n-2)*f(n-1); fi; end;
%p [seq(f(n), n=0..20)]; # This is for the unsigned version. - _N. J. A. Sloane_, May 09 2016
%p seq(simplify((-1)^n*KummerU(-n, -2*n, -1)), n = 0..17); # _Peter Luschny_, May 10 2022
%t Table[(-1)^k (2k)! Hypergeometric1F1[-k, -2k, -1]/k!, {k, 0, 10}] (* _Vladimir Reshetnikov_, Feb 16 2011 *)
%t nxt[{n_,a_,b_}]:={n+1,b,a-2b(2n+1)}; NestList[nxt,{1,1,-1},20][[All,2]] (* _Harvey P. Dale_, Aug 18 2017 *)
%o (PARI) {a(n)= if(n<0, n=-n-1); sum(k=0, n, (2*n-k)!/ (k!*(n-k)!)* (-1)^(n-k) )} /* _Michael Somos_, Apr 02 2007 */
%o (PARI) {a(n)= local(A); if(n<0, n= -n-1); A= sqrt(1 +4*x +x*O(x^n)); n!*polcoeff( exp((A-1)/2)/A, n)} /* _Michael Somos_, Apr 02 2007 */
%o (PARI) {a(n)= local(A); if(n<0, n= -n-1); n+=2 ; for(k= 1, n, A+= x*O(x^k); A= truncate( (1+x)* exp(A) -1-A) ); A+= x*O(x^n); A-= A^2; -(-1)^n*n!* polcoeff( serreverse(A), n)} /* _Michael Somos_, Apr 02 2007 */
%o (Sage)
%o A002119 = lambda n: hypergeometric([-n, n+1], [], 1)
%o [simplify(A002119(n)) for n in (0..17)] # _Peter Luschny_, Oct 17 2014
%Y Cf. A053556, A053557, A001514, A065920, A065921, A065922, A065707, A000806, A006199, A065923.
%Y See also A033815.
%Y Numerators of the convergents of e are A001517, which has a similar interpretation to a(n) in terms of monomer-dimer configurations, but omitting the restriction that adjacent monomers not be mapped to each other by the bijection.
%Y Polynomial coefficients are in A001498.
%K sign,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Apr 03 2000
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