OFFSET
0,3
LINKS
Lars Blomberg and Gheorghe Coserea, Table of n, a(n) for n = 0..777, terms 1..100 from Lars Blomberg.
Juan S. Auli and Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020.
Sylvie Corteel, Megan A. Martinez, Carla D. Savage, and Michael Weselcouch, Patterns in Inversion Sequences I, arXiv:1510.05434 [math.CO], 2015. See equations (4,5).
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
Jay Pantone, The enumeration of inversion sequences avoiding the patterns 201 and 210, arXiv:2310.19632 [math.CO], 2023.
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024. See p. 2.
FORMULA
a(n) = Sum_{k=0..n-1} Sum_{p=-1,k-1} T(n,k,p), where T(n,k,p) = Sum_{i=-1..p} T(n-1,k,i) + Sum_{j=p+1..k} T(n-1,j,p) with initial conditions T(n,k,p) = 0 if k >= n and T(n,k,-1) = (n-k)/n * binomial(n-1+k,k). (eqn. (4) and (5) in Corteel link) - Gheorghe Coserea, Sep 21 2017
a(n) ~ c * (27/2)^n / n^alfa, where alfa = 5.7667921227... and c = 9.973... - Vaclav Kotesovec, Oct 16 2021
MATHEMATICA
T[n_, k_, _] /; k >= n = 0; T[n_, k_, -1] := (n-k)/n*Binomial[n+k-1, k];
T[n_, k_, p_] := T[n, k, p] = Sum[T[n-1, k, i], {i, -1, p}] + Sum[T[n-1, j, p], {j, p+1, k}];
a[0] = 1; a[n_] := Sum[T[n, k, p], {k, 0, n-1}, {p, -1, k-1}];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Aug 10 2018 *)
PROG
(PARI)
seq(N) = {
my(a=vector(N), t=vector(2, k, matrix(N, N)), s=matrix(N+1, N+1),
C=(n, k)->(n-k)/n*binomial(n-1+k, k));
for (n=1, N, for (k=1, n, for(p=1, k-1,
s[k+1, p+1] = s[k+1, p] + t[1+n%2][k, p];
s[p+1, k+1] = s[p+1, k] + t[1+n%2][k, p];
t[1+(n+1)%2][k, p] = s[k+1, p+1] + s[p+1, k+1] + C(n-1, k-1)));
a[n] = sum(k=1, n, sum(p=1, k-1, t[1+(n+1)%2][k, p])) + C(n+1, n));
a;
};
concat(1, seq(23)) \\ Gheorghe Coserea, Nov 20 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Oct 26 2015
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Dec 15 2016
More terms from Lars Blomberg, Jan 18 2017
STATUS
approved