A002119 Bessel polynomial y_n(-2).
1, -1, 7, -71, 1001, -18089, 398959, -10391023, 312129649, -10622799089, 403978495031, -16977719590391, 781379079653017, -39085931702241241, 2111421691000680031, -122501544009741683039, 7597207150294985028449, -501538173463478753560673
Offset: 0
Examples
Example from _William P. Orrick_, Jan 19 2023: (Start)
For n=2 the Bessel polynomial is y_2(x) = 1 + 3x + 3x^2 which satisfies y_2(-2) = -7.
The |a(2)|=7 dice pairs are
{{0,1,2,3,4,5}, {0,6,12,18,24,30}},
{{0,1,2,18,19,20}, {0,3,6,9,12,15}},
{{0,1,2,9,10,11}, {0,3,6,18,21,24}},
{{0,1,6,7,12,13}, {0,2,4,18,20,22}},
{{0,1,12,13,24,25}, {0,2,4,6,8,10}},
{{0,1,2,6,7,8}, {0,3,12,15,24,27}},
{{0,1,4,5,8,9}, {0,2,12,14,24,26}}.
The corresponding Krasner factorizations of (x^36-1)/(x-1) are
{(x^6-1)/(x-1), (x^36-1)/(x^6-1)},
{((x^36-1)/(x^18-1))*((x^3-1)/(x-1)), (x^18-1)/(x^3-1)},
{((x^18-1)/(x^9-1))*((x^3-1)/(x-1)), ((x^36-1)/(x^18-1))*((x^9-1)/(x^3-1))},
{((x^18-1)/(x^6-1))*((x^2-1)/(x-1)), ((x^36-1)/(x^18-1))*((x^6-1)/(x^2-1))},
{((x^36-1)/(x^12-1))*((x^2-1)/(x-1)), (x^12-1)/(x^2-1)},
{((x^12-1)/(x^6-1))*((x^3-1)/(x-1)), ((x^36-1)/(x^12-1))*((x^6-1)/(x^3-1))},
{((x^12-1)/(x^4-1))*((x^2-1)/(x-1)), ((x^36-1)/(x^12-1))*((x^4-1)/(x^2-1))}.
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
(==, {}),
(B=R, {(3,1)}),
(BBRR, {(3,1),(4,2)}),
(RBBR, {(1,3),(4,2)}),
(R=B, {(1,3)}),
(BRRB, {(2,4),(3,1)}),
(RRBB, {(1,3),(2,4)}).
(End)
References
- L. Euler, 1737.
- 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.
- J. W. L. Glaisher, On Lambert's proof of the irrationality of Pi and on the irrationality of certain other quantities, Reports of British Assoc. Adv. Sci., 1871, pp. 16-18.
- M. Krasner and B. Ranulac, Sur une propriété des polynomes de la division du cercle, Comptes Rendus Académie des Sciences Paris, 240:397-399, 1937.
- 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
Crossrefs
Programs
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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 seq(simplify((-1)^n*KummerU(-n, -2*n, -1)), n = 0..17); # Peter Luschny, May 10 2022
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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 *) -
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
Formula
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
Extensions
More terms from Vladeta Jovovic, Apr 03 2000
Comments