OFFSET
1,2
COMMENTS
a(n) = x(n) + y(n) where x(n)/y(n) is the continued fraction [1,2,3,4,...,n].
Using a(n) = x(n) - y(n) instead of a(n) = x(n) + y(n) would give A058307.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..449
Juan S. Auli and Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020.
Zhicong Lin, Sherry H. F. Yan, Vincular patterns in inversion sequences, Applied Mathematics and Computation (2020), Vol. 364, 124672.
FORMULA
From Paul D. Hanna, Oct 31 2006: (Start)
a(n+1) = Sum_{k=0..n} k!*C(floor((n+k)/2),k)*C(floor((n+k+1)/2),k).
a(n+1) = Sum_{k=0..n} k!*A124428(n+k,k). (End)
MATHEMATICA
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 *)
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 *)
PROG
(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
(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
(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
(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
CROSSREFS
KEYWORD
base,easy,nonn
AUTHOR
Eric Angelini, Feb 12 2005
EXTENSIONS
Edited and extended by Robert G. Wilson v, Feb 14 2005
STATUS
approved