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A058797
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a(n) = n*a(n-1) - a(n-2), with a(-1) = 0, a(0) = 1.
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16
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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
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OFFSET
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-1,5
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COMMENTS
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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
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REFERENCES
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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]
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LINKS
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FORMULA
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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
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
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EXAMPLE
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G.f. = 1 + x + x^2 + 2*x^3 + 7*x^4 + 33*x^5 + 191*x^6 + ... - Michael Somos, Mar 10 2023
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MAPLE
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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)
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MATHEMATICA
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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 *)
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PROG
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(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
if n==-1: return 0
elif n==0: return 1
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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More terms from Tyrrell B. McAllister (tmcal(AT)math.ucdavis.edu), May 05 2003
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STATUS
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approved
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