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A058797 a(-1) = 0, a(0) = 1; thereafter a(n) = n*a(n-1) - a(n-2). 11
0, 1, 1, 1, 2, 7, 33, 191, 1304, 10241, 90865, 898409, 9791634, 116601199, 1506023953, 20967734143, 313009988192, 4987192076929, 84469255319601, 1515459403675889, 28709259414522290, 572669728886769911, 11997355047207645841 (list; graph; refs; listen; history; text; internal format)
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

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. - Tyrrell B. McAllister (tmcal(AT)math.ucdavis.edu), May 05 2003

(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}^infty(-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 Book, New York, 1945, page 144 [From Roger L. Bagula and Bob Hanlon (hanlonr(AT)cox.net), Sep 03 2006]

Lehmer, D. H., "Arithmetical periodicities of Bessel functions." Annals of Mathematics, 33 (1932): 143-150. The sequence is on page 150.

LINKS

Harry J. Smith, Table of n, a(n) for n = -1..200

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 integer n. - Michael Somos, Mar 07 2011

a(-2-n) = -(-1)^n * a(n). - Michael Somos, Mar 07 2011

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)

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) = local(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 */

CROSSREFS

Column 0 of A007754.

Cf. A001053, A106174, A084950 (alternating row sums).

Sequence in context: A186760 A162661 A104981 * A121965 A006595 A179525

Adjacent sequences:  A058794 A058795 A058796 * A058798 A058799 A058800

KEYWORD

nonn

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

Corrected typo in definition in Name line. Corrected typo in cross-reference: A001503 -> A001053 (twice). - Svante Janson, Nov 24 2008

STATUS

approved

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Last modified March 29 07:21 EDT 2017. Contains 284250 sequences.