OFFSET
-1,5
COMMENTS
a(n) is also the determinant of the symmetric, tridiagonal n X n matrix with entries equal 1 just above and below the diagonal and diagonal entries 1, 2, .., n. Example: a(4)=det(matrix([[1, 1, 0, 0], [1, 2, 1, 0], [0, 1, 3, 1], [0, 0, 1, 4]])). - Roland Bacher, Jun 19 2001
From Tyrrell B. McAllister (tmcal(AT)math.ucdavis.edu), May 05 2003: (Start)
For n >= 1, a(n+1) counts the Gelfand-Tsetlin patterns x = (x_{ij})_{1 <= i <= j <= n} (i.e., triangular arrays such that x_{ij} >= 0 for 1 <= i <= j <= n and x_{i,j+1} >= x_{ij} >= x_{i+1,j+1} for 1 <= i <= j <= n-1) that satisfy the additional conditions that
- all the entries of x are integral,
- x_{nn} = x_{n-1,n-1} = 0,
- x_{ij} - x{i+1,j+1} <= 1, for 1 <= i <= j <= n-1,
- x_{in}-1 <= x_{ii} <= x_{i+1,n}, for 1 <= i <= n-1. (End)
(a(n), n >= 1) is the Hankel transform of the Bessel numbers (A006789) starting at n=1. Example: a(3) = det({{1, 2, 5}, {2, 5, 14}, {5, 14, 43}}) = 2. - David Callan, Nov 29 2007
a(n) is the number of permutations of [n] in which each descent is the 32 of a 1-32 pattern or the 21 of a 3-21 pattern or both. (These are generalized patterns where a dash between two entries means the corresponding permutation entries do not have to be adjacent and the absence of a dash means they do.) Example: 3462175 fails the condition because 62 is a descent and no entry preceding the 6 lies outside the interval [2,6]; a(4)=7 counts 1234, 1243, 1324, 1342, 1423, 1432, 2431. Outline of proof: Partition the permutations counted by a(n) according to their last entry. The number of permutations with last entry 1 is a(n-1)-a(n-2) and, for 2 <= k <= n, the number with last entry k is a(n-1). These observations give Bottomley's recurrence below. - David Callan, Jul 22 2008
Starting with offset 1 = eigensequence of an infinite lower triangular matrix with (1, 2, 3, ...) as the main diagonal and (-1, -1, -1, ...) as the subdiagonal. - Gary W. Adamson, Apr 20 2009
a(n) is the denominator sequence for the n-th approximation of the continued fraction -(0 + K_{k>=1}(-1/k)) = 1/(1-1/(2-1/(3-1/4-... The corresponding numerator sequence is A058798(n). The limit is BesselJ(1,2)/BesselJ(0,2) = 2.575920321... See A084950 for a comment on asymptotics of Bessel functions. See also the limit a(n)/n! given in the first line of the formula section, and of A058798(n)/n! given as a formula there. - Wolfdieter Lang, Mar 08 2013
REFERENCES
Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Publications, New York, 1945, page 144. [Roger L. Bagula and Bob Hanlon (hanlonr(AT)cox.net), Sep 03 2006]
LINKS
Harry J. Smith, Table of n, a(n) for n = -1..200
D. H. Lehmer, Arithmetical periodicities of Bessel functions, Annals of Mathematics, 33 (1932): 143-150. The sequence is on page 150.
Eric Weisstein's World of Mathematics, Bessel Function of the First Kind
FORMULA
a(n) is asymptotic to c*n! with c = BesselJ(0, 2) = Sum (-1)^k/(k!)^2 = 0.223890779... (A091681). - Franklin T. Adams-Watters and Alec Mihailovs (alec(AT)mihailovs.com), Aug 17 2005
a(n) = n*a(n-1) - a(n-2) [with a(0) = 1 and a(-1) = 0] = A058798(n-1) - A058799(n-2). - Henry Bottomley, Feb 28 2001
E.g.f.: Pi*(BesselY(0, -2)*BesselJ(1, 2*sqrt(1-x))+BesselJ(0, 2)*BesselY(1, -2*sqrt(1-x)))/sqrt(1-x). - Alec Mihailovs (alec(AT)mihailovs.com), Aug 20 2005
a(n) = Pi*(BesselJ[n+1, 2]*BesselY[0, 2] - BesselJ[0, 2]*BesselY[n+1, 2]). - Roger L. Bagula, Sep 03 2006. [Offset adapted. - Wolfdieter Lang, Mar 08 2013]
If b(n) = a(n-1) / a(n), then b(n) = 1 / (n - b(n-1)) unless n=0 or n=-1. - Michael Somos, Mar 07 2011
a(n+2) * (a(n) + a(n+1) + a(n+2)) = a(n+1) * (a(n+1) + a(n+3)) for all n in Z. - Michael Somos, Mar 07 2011
a(-2-n) = -(-1)^n * a(n) for all n in Z. - Michael Somos, Mar 07 2011
a(n) = Sum_{k = 0..floor((n-1)/2)} (-1)^k * binomial(n-k-1, k)*(n-k-1)!/k!. Cf. A058798. - Peter Bala, Feb 12 2024
EXAMPLE
G.f. = 1 + x + x^2 + 2*x^3 + 7*x^4 + 33*x^5 + 191*x^6 + ... - Michael Somos, Mar 10 2023
MAPLE
A058797:=rsolve({a(n) = n*a(n-1)-a(n-2), a(0)=1, a(1)=1}, a(n), makeproc); # Alec Mihailovs (alec(AT)mihailovs.com)
MATHEMATICA
RecurrenceTable[{a[n]==n*a[n-1]-a[n-2], a[-1]==0, a[0]==1}, a, {n, -1, 30}] (* G. C. Greubel, Nov 24 2018 *)
PROG
(PARI) { a1=0; a2=1; f="b058797.txt"; write(f, "-1 0"); write(f, "0 1"); for (n=1, 200, a=n*a2-a1; a1=a2; a2=a; write(f, n, " ", a); ); } /* Harry J. Smith, Jun 23 2009 */
(PARI) {a(n) = my(s, a0, a1, a2); n++; s = sign(n); s^n * if( n!=0, a1 = 1; for( k=2, abs(n), a2 = (k-1) * a1 - a0; a0 = a1; a1 = a2); a1)}; /* Michael Somos, Mar 07 2011 */
(Magma) [0, 1] cat [n le 2 select 1 else n*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 28 2017
(Sage)
@cached_function
def A058797(n):
if n==-1: return 0
elif n==0: return 1
[A058797(n) for n in (-1..30)] # G. C. Greubel, Nov 24 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Christian G. Bower, Dec 02 2000
EXTENSIONS
More terms from Tyrrell B. McAllister (tmcal(AT)math.ucdavis.edu), May 05 2003
Edited by N. J. A. Sloane, Sep 25 2008, at the suggestion of Christopher Carl Heckman
Typo in name corrected by Svante Janson, Nov 24 2008
STATUS
approved