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A263777 Number of inversion sequences avoiding pattern 201 (or 210).

%I A263777 #53 Jul 15 2024 10:37:59
%S A263777 1,1,2,6,24,118,674,4306,29990,223668,1763468,14558588,124938648,
%T A263777 1108243002,10115202962,94652608690,905339525594,8829466579404,
%U A263777 87618933380020,883153699606024,9028070631668540,93478132393544988,979246950529815364,10368459385853924212
%N A263777 Number of inversion sequences avoiding pattern 201 (or 210).
%H A263777 Lars Blomberg and Gheorghe Coserea, <a href="/A263777/b263777.txt">Table of n, a(n) for n = 0..777</a>, terms 1..100 from Lars Blomberg.
%H A263777 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 A263777 Sylvie Corteel, Megan A. Martinez, Carla D. Savage, and Michael Weselcouch, <a href="http://arxiv.org/abs/1510.05434">Patterns in Inversion Sequences I</a>, arXiv:1510.05434 [math.CO], 2015. See equations (4,5).
%H A263777 Megan A. Martinez and Carla D. Savage, <a href="https://arxiv.org/abs/1609.08106">Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations</a>, arXiv:1609.08106 [math.CO], 2016.
%H A263777 Jay Pantone, <a href="https://arxiv.org/abs/2310.19632">The enumeration of inversion sequences avoiding the patterns 201 and 210</a>, arXiv:2310.19632 [math.CO], 2023.
%H A263777 Benjamin Testart, <a href="https://arxiv.org/abs/2407.07701">Completing the enumeration of inversion sequences avoiding one or two patterns of length 3</a>, arXiv:2407.07701 [math.CO], 2024. See p. 2.
%F A263777 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
%F A263777 a(n) ~ c * (27/2)^n / n^alfa, where alfa = 5.7667921227... and c = 9.973... - _Vaclav Kotesovec_, Oct 16 2021
%t A263777 T[n_, k_, _] /; k >= n = 0; T[n_, k_, -1] := (n-k)/n*Binomial[n+k-1, k];
%t A263777 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}];
%t A263777 a[0] = 1; a[n_] := Sum[T[n, k, p], {k, 0, n-1}, {p, -1, k-1}];
%t A263777 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Aug 10 2018 *)
%o A263777 (PARI)
%o A263777 seq(N) = {
%o A263777   my(a=vector(N), t=vector(2, k, matrix(N, N)), s=matrix(N+1,N+1),
%o A263777      C=(n,k)->(n-k)/n*binomial(n-1+k, k));
%o A263777   for (n=1, N, for (k=1, n, for(p=1, k-1,
%o A263777       s[k+1,p+1] = s[k+1,p] + t[1+n%2][k,p];
%o A263777       s[p+1,k+1] = s[p+1,k] + t[1+n%2][k,p];
%o A263777       t[1+(n+1)%2][k, p] = s[k+1,p+1] + s[p+1,k+1] + C(n-1,k-1)));
%o A263777     a[n] = sum(k=1, n, sum(p=1, k-1, t[1+(n+1)%2][k, p])) + C(n+1, n));
%o A263777   a;
%o A263777 };
%o A263777 concat(1, seq(23)) \\ _Gheorghe Coserea_, Nov 20 2017
%Y A263777 Cf. A263778, A263779, A263780.
%K A263777 nonn
%O A263777 0,3
%A A263777 _Michel Marcus_, Oct 26 2015
%E A263777 a(0)=1 prepended by _Alois P. Heinz_, Dec 15 2016
%E A263777 More terms from _Lars Blomberg_, Jan 18 2017