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A201692 Number of permutations that avoid the consecutive pattern 1423.

%I A201692 #41 Oct 27 2023 21:38:02
%S A201692 1,1,2,6,23,110,631,4218,32221,276896,2643883,27768955,318174363,
%T A201692 3949415431,52794067318,756137263377,11551672922816,187507250145806,
%U A201692 3222662529113641,58464560588277289,1116469710152742025,22386721651323946628,470259350616967829363
%N A201692 Number of permutations that avoid the consecutive pattern 1423.
%H A201692 Alois P. Heinz, <a href="/A201692/b201692.txt">Table of n, a(n) for n = 0..250</a> (terms n = 0..60 from Ray Chandler)
%H A201692 A. Baxter, B. Nakamura, and D. Zeilberger. <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/auto.html">Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes</a>
%H A201692 V. Dotsenko and A. Khoroshkin, <a href="http://arxiv.org/abs/1109.2690">Shuffle algebras, homology, and consecutive pattern avoidance</a>, arXiv preprint arXiv:1109.2690, 2011
%F A201692 The reference gives an e.g.f. There is an associated triangle of numbers c_{n,l} that should be added to the OEIS if it is not already present.
%F A201692 a(n) ~ c * d^n * n!, where d = 0.95482605094987833345080179991528996596888600981..., c = 1.1567436851576902067739566662625378535625602... . - _Vaclav Kotesovec_, Sep 11 2014
%p A201692 c := proc(n,l)
%p A201692     if n = 1 then
%p A201692         if l = 0 then
%p A201692             1;
%p A201692         else
%p A201692             0;
%p A201692         end if;
%p A201692     elif n= 2 or n = 3 then
%p A201692         0;
%p A201692     else
%p A201692         a := 0 ;
%p A201692         for k from 1 to (n-2)/2 do
%p A201692             a := a+procname(n-2*k-1,l-k)*binomial(n-k-2,k) ;
%p A201692         end do:
%p A201692         a ;
%p A201692     end if;
%p A201692 end proc:
%p A201692 A201693 := proc(nmax)
%p A201692     g := 1-t ;
%p A201692     for n from 2 to nmax do
%p A201692         for l from 0 to n/2 do
%p A201692             g := g-c(n,l)*t^n*(-1)^l/n! ;
%p A201692         end do:
%p A201692     end do:
%p A201692     taylor(1/g,t=0,nmax) ;
%p A201692 end proc:
%p A201692 nmax := 25 ;
%p A201692 egf := A201693(nmax) ;
%p A201692 for n from 0 to nmax-1 do
%p A201692     printf("%d,",coeftayl(egf,t=0,n)*n!) ;
%p A201692 end do: # _R. J. Mathar_, Dec 04 2011
%p A201692 # second Maple program:
%p A201692 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A201692       add(b(u-j, o+j-1, `if`(0<t and j<t, -j, 0)), j=1..u)+
%p A201692       add(b(u+j-1, o-j, j), j=`if`(t<0, -t, 1)..o))
%p A201692     end:
%p A201692 a:= n-> b(n, 0$2):
%p A201692 seq(a(n), n=0..25);  # _Alois P. Heinz_, Nov 07 2013
%t A201692 b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[b[u-j, o+j-1, If[0<t && j<t, -j, 0]], {j, 1, u}] + Sum[b[u+j-1, o-j, j], {j, If[t<0, -t, 1], o}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Mar 18 2014, after _Alois P. Heinz_ *)
%Y A201692 Cf. A113228, A113229, A117156, A117158, A117226, A201693.
%K A201692 nonn,easy
%O A201692 0,3
%A A201692 _N. J. A. Sloane_, Dec 03 2011
%E A201692 Definition corrected by _N. J. A. Sloane_, Mar 15 2015