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A058797
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a(-1) = 0, a(0) = 1; thereafter a(n) = n*a(n-1)-a(n-2).
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5
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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; internal format)
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OFFSET
| -1,5
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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 (Roland.Bacher(AT)ujf-grenoble.fr), 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 (callan(AT)stat.wisc.edu), 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 (callan(AT)stat.wisc.edu), Jul 22 2008
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
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 20 2009: (Start)
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. (End)
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REFERENCES
| 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]
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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
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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, 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 (se16(AT)btinternet.com), 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, 2]*BesselY[0, 2] - BesselJ[0, 2]*BesselY[n, 2]). - Roger Bagula and Bob Hanlon (rlbagulatftn(AT)yahoo.com), Sep 03 2006
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
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MAPLE
| A058797:=rsolve({a(n) = n*a(n-1)-a(n-2), a(0)=1, a(1)=1}, a(n), makeproc); (Mihailovs)
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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); ); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), 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 */
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CROSSREFS
| Column 0 of A007754.
Cf. A001053, A106174.
Sequence in context: A162661 A104981 A121965 * A006595 A179525 A059099
Adjacent sequences: A058794 A058795 A058796 * A058798 A058799 A058800
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KEYWORD
| nonn
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net), Dec 02 2000
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EXTENSIONS
| More terms from Tyrrell B. McAllister (tmcal(AT)math.ucdavis.edu), May 05 2003
Edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 25 2008, at the suggestion of Christopher Heckman.
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
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