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A102038 a(n+1) = n*a(n) + a(n-1), a(1)=1, a(2)=2.

%I A102038 #38 Jan 08 2025 12:49:14
%S A102038 1,2,5,17,73,382,2365,16937,137861,1257686,12714721,141119617,
%T A102038 1706150125,22321071242,314201147513,4735338283937,76079613690505,
%U A102038 1298088771022522,23441677492095901,446689961120844641
%N A102038 a(n+1) = n*a(n) + a(n-1), a(1)=1, a(2)=2.
%C A102038 a(n) = x(n) + y(n) where x(n)/y(n) is the continued fraction [1,2,3,4,...,n].
%C A102038 Using a(n) = x(n) - y(n) instead of a(n) = x(n) + y(n) would give A058307.
%H A102038 G. C. Greubel, <a href="/A102038/b102038.txt">Table of n, a(n) for n = 1..449</a>
%H A102038 Juan S. Auli and Sergi Elizalde, <a href="https://arxiv.org/abs/2003.11533">Wilf equivalences between vincular patterns in inversion sequences</a>, arXiv:2003.11533 [math.CO], 2020.
%H A102038 Zhicong Lin and Sherry H. F. Yan, <a href="https://doi.org/10.1016/j.amc.2019.124672">Vincular patterns in inversion sequences</a>, Applied Mathematics and Computation (2020), Vol. 364, 124672.
%H A102038 Shuzhen Lv and Philip B. Zhang, <a href="https://arxiv.org/abs/2501.00357">Joint equidistributions of mesh patterns 123 and 321 with symmetric and antipodal shadings</a>, arXiv:2501.00357 [math.CO], 2024. See p. 17.
%F A102038 a(n) = A001040(n) + A001053(n) for n>1.
%F A102038 From _Paul D. Hanna_, Oct 31 2006: (Start)
%F A102038 a(n+1) = Sum_{k=0..n} k!*C(floor((n+k)/2),k)*C(floor((n+k+1)/2),k).
%F A102038 a(n+1) = Sum_{k=0..n} k!*A124428(n+k,k). (End)
%t A102038 a[1]= 1; a[2]= 2; a[n_]:= a[n] = (n-1)*a[n-1]+a[n-2]; Table[a[n], {n,20}] (* _Robert G. Wilson v_, Feb 14 2005 *)
%t A102038 RecurrenceTable[{a[1]==1,a[2]==2,a[n+1]==n*a[n]+a[n-1]},a,{n,20}] (* _Harvey P. Dale_, Sep 04 2018 *)
%o A102038 (PARI) a(n)=sum(k=0,n,k!*binomial((n+k)\2,k)*binomial((n+k+1)\2,k)) \\ _Paul D. Hanna_, Oct 31 2006
%o A102038 (Magma) I:=[1,2]; [n le 2 select I[n] else (n-1)*Self(n-1) +Self(n-2): n in [1..30]]; // _G. C. Greubel_, Feb 23 2019
%o A102038 (Sage) [sum(factorial(k)*binomial(floor((n+k-1)/2), k)*binomial(floor((n+k)/2), k) for k in (0..n)) for n in (1..30)] # _G. C. Greubel_, Feb 23 2019
%o A102038 (GAP) a:=[1,2];; for n in [3..30] do a[n]:=(n-1)*a[n-1]+a[n-2]; od; a; # _G. C. Greubel_, Feb 23 2019
%Y A102038 Numerators are in A001040 and denominators in A001053.
%Y A102038 Cf. A124428.
%K A102038 base,easy,nonn
%O A102038 1,2
%A A102038 _Eric Angelini_, Feb 12 2005
%E A102038 Edited and extended by _Robert G. Wilson v_, Feb 14 2005