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A264896 Number of permutations of [n] with exactly one occurrence of the consecutive pattern 45321.

%I A264896 #11 Nov 01 2021 04:23:23
%S A264896 1,12,126,1344,15110,180736,2308548,31481472,457520055,7068885600,
%T A264896 115808906178,2006533573632,36674815572540,705453732298240,
%U A264896 14248697953325160,301564509817810944,6674811622886359005,154228999030804811520,3713903962096887036390
%N A264896 Number of permutations of [n] with exactly one occurrence of the consecutive pattern 45321.
%C A264896 Consecutive patterns 12354, 21345, 54312 give the same sequence.
%H A264896 Alois P. Heinz, <a href="/A264896/b264896.txt">Table of n, a(n) for n = 5..200</a>
%e A264896 a(5) = 1: 45321.
%e A264896 a(6) = 12: 156432, 256431, 356421, 453216, 456321, 463215, 546321, 563214, 564213, 564312, 564321, 645321.
%p A264896 b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(
%p A264896        b(u+j-1, o-j, `if`(u+j-3<j, 0, j)), j=1..o)+convert(series(
%p A264896       `if`(t=-2, x, 1)*add(b(u-j, o+j-1, `if`(j<t or t=-2, 0,
%p A264896       `if`(t>0, -1, `if`(t=-1, -2, 0)))), j=1..u),x,2),polynom))
%p A264896     end:
%p A264896 a:= n-> coeff(b(n, 0$2),x,1):
%p A264896 seq(a(n), n=5..25);
%t A264896 b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Sum[b[u + j - 1, o - j, If[u + j - 3 < j, 0, j]], {j, 1, o}] + Expand[If[t == -2, x, 1]*Sum[b[u - j, o + j - 1, If[j < t || t == -2, 0, If[t > 0, -1, If[t == -1, -2, 0]]]], {j, 1, u}]]];
%t A264896 a[n_] := Coefficient[b[n, 0, 0], x, 1];
%t A264896 Table[a[n], {n, 5, 25}] (* _Jean-François Alcover_, Nov 01 2021, after _Alois P. Heinz_ *)
%Y A264896 Column k=1 of A264781.
%K A264896 nonn
%O A264896 5,2
%A A264896 _Alois P. Heinz_, Nov 27 2015