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A002119
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Bessel polynomial y_n(-2).
(Formerly M4444 N1880)
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16
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1, -1, 7, -71, 1001, -18089, 398959, -10391023, 312129649, -10622799089, 403978495031, -16977719590391, 781379079653017, -39085931702241241, 2111421691000680031, -122501544009741683039, 7597207150294985028449, -501538173463478753560673
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OFFSET
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0,3
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COMMENTS
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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...)))))).
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
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REFERENCES
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L. Euler, 1737.
J. W. L. Glaisher, Reports of British Assoc. Adv. Sci., 1871, pp. 16-18.
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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T. D. Noe, Table of n, a(n) for n = 0..100
Leo Chao, Paul DesJarlais and John L Leonard, A binomial identity, via derangements, Math. Gaz. 89 (2005), 268-270.
D. H. Lehmer, Arithmetical periodicities of Bessel functions, Annals of Mathematics, 33 (1932): 143-150. The sequence but with all signs positive is on page 149.
D. H. Lehmer, Review of various tables by P. Pederson, Math. Comp., 2 (1946), 68-69.
Index entries for sequences related to Bessel functions or polynomials
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FORMULA
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D-finite with recurrence a(n) = -2(2n-1)*a(n-1) + a(n-2). - T. D. Noe, Oct 26 2006
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
a(-n-1) = a(n). - Michael Somos, Apr 02 2007
a(n) = (1/n!)*Integral_{x>=-1} (-x*(1+x))^n*exp(-(1+x)). - Paul Barry, Apr 19 2010
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
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
a(n) = hypergeom([-n, n+1], [], 1). - Peter Luschny, Oct 17 2014
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
a(n) ~ (-1)^n * 2^(2*n+1/2) * n^n / exp(n+1/2). - Vaclav Kotesovec, Jul 22 2015
From G. C. Greubel, Aug 16 2017: (Start)
G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; -4*t/(1-t)^2).
E.g.f.: (1+4*t)^(-1/2) * exp((sqrt(1+4*t) - 1)/2). (End)
a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*binomial(n+k,k)*k!. - Ilya Gutkovskiy, Nov 24 2017
a(n) = (-1)^n*KummerU(-n, -2*n, -1). - Peter Luschny, May 10 2022
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MAPLE
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f:=proc(n) option remember; if n <= 1 then 1 else f(n-2)+(4*n-2)*f(n-1); fi; end;
[seq(f(n), n=0..20)]; # This is for the unsigned version. - N. J. A. Sloane, May 09 2016
seq(simplify((-1)^n*KummerU(-n, -2*n, -1)), n = 0..17); # Peter Luschny, May 10 2022
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MATHEMATICA
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Table[(-1)^k (2k)! Hypergeometric1F1[-k, -2k, -1]/k!, {k, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)
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 *)
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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)^(n-k) )} /* Michael Somos, Apr 02 2007 */
(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 */
(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 */
(Sage)
A002119 = lambda n: hypergeometric([-n, n+1], [], 1)
[simplify(A002119(n)) for n in (0..17)] # Peter Luschny, Oct 17 2014
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CROSSREFS
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Cf. A001517, A053556, A053557, A001514, A065920, A065921, A065922, A065707, A000806, A006199, A065923.
See also A033815.
Polynomial coefficients are in A001498.
Sequence in context: A334135 A268702 A052390 * A146752 A022518 A113053
Adjacent sequences: A002116 A002117 A002118 * A002120 A002121 A002122
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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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EXTENSIONS
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More terms from Vladeta Jovovic, Apr 03 2000
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STATUS
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approved
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