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A002119 Bessel polynomial y_n(-2).
(Formerly M4444 N1880)
13
1, -1, 7, -71, 1001, -18089, 398959, -10391023, 312129649, -10622799089, 403978495031, -16977719590391, 781379079653017, -39085931702241241, 2111421691000680031, -122501544009741683039, 7597207150294985028449, -501538173463478753560673 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

REFERENCES

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).

LINKS

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

FORMULA

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

MAPLE

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A001517, A053556, A053557, A001514, A065920, A065921, A065922, A065707, A000806, A006199, A065923.

See also A033815.

Polynomial coefficients are in A001498.

Sequence in context: A067307 A268702 A052390 * A146752 A022518 A113053

Adjacent sequences:  A002116 A002117 A002118 * A002120 A002121 A002122

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Apr 03 2000

STATUS

approved

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Last modified October 18 03:25 EDT 2019. Contains 328135 sequences. (Running on oeis4.)