%I A058797
%S A058797 0,1,1,1,2,7,33,191,1304,10241,90865,898409,9791634,116601199,1506023953,
%T A058797 20967734143,313009988192,4987192076929,84469255319601,
%U A058797 1515459403675889,28709259414522290,572669728886769911
%N A058797 a(-1) = 0, a(0) = 1; thereafter a(n) = n*a(n-1)-a(n-2).
%C A058797 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 (Roland.Bacher(AT)ujf-grenoble.fr), 
               Jun 19 2001
%C A058797 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
%C A058797 (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 (callan(AT)stat.wisc.edu), Nov 29 2007
%C A058797 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 (callan(AT)stat.wisc.edu), Jul 22 2008
%C A058797 An improved version of the Bessel-Binet recursion derived from Z(p-1)+Z(p+1)=(2*p)/
               x)*Z(p) that is A001053-like. - Roger Bagula and Bob Hanlon (rlbagulatftn(AT)yahoo.com), 
               Sep 03 2006
%C A058797 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 20 2009: 
               (Start)
%C A058797 Starting with offset 1 = eigensequence of an infinite lower triangular 
               matrix
%C A058797 with (1, 2, 3,...) as the main diagonal and (-1, -1, -1,...) as the subdiagonal. 
               (End)
%D A058797 Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, 
               Dover Book, New York,1945, page144 [From Roger Bagula and Bob Hanlon 
               (rlbagulatftn(AT)yahoo.com), Sep 03 2006]
%H A058797 Harry J. Smith, <a href="b058797.txt">Table of n, a(n) for n=-1,...,200</
               a>
%H A058797 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BesselFunctionoftheFirstKind.html">Bessel Function of the First Kind</
               a>
%F A058797 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, 
               Aug 17 2005)
%F A058797 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 (se16(AT)btinternet.com), Feb 28 2001
%F A058797 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
%F A058797 a(n) = Pi*(BesselJ[n, 2]*BesselY[0, 2] - BesselJ[0, 2]*BesselY[n, 2]). 
               - Roger Bagula and Bob Hanlon (rlbagulatftn(AT)yahoo.com), Sep 03 
               2006
%p A058797 A058797:=rsolve({a(n) = n*a(n-1)-a(n-2),a(0)=1,a(1)=1},a(n),makeproc); 
               (Mihailovs)
%o A058797 (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); ); } 
               [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jun 23 2009]
%Y A058797 Column 0 of A007754.
%Y A058797 Cf. A001053, A106174.
%Y A058797 Sequence in context: A086618 A162661 A104981 this_sequence A006595 A059099 
               A020103
%Y A058797 Adjacent sequences: A058794 A058795 A058796 this_sequence A058798 A058799 
               A058800
%K A058797 nonn
%O A058797 -1,5
%A A058797 Christian G. Bower (bowerc(AT)usa.net), Dec 02 2000
%E A058797 More terms from Tyrrell B. McAllister (tmcal(AT)math.ucdavis.edu), May 
               05 2003
%E A058797 Edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 25 2008, at 
               the suggestion of Christopher Heckman.
%E A058797 Corrected typo in definition in Name line. Corrected typo in cross-reference: 
               A001503 -> A001053 (twice). - Svante Janson (svante.janson(AT)math.uu.se), 
               Nov 24 2008