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A231386 Number of permutations of [n] with exactly one occurrence of one of the consecutive step patterns UUD, UDU, DUU (U=up, D=down).

%I A231386 #13 Dec 22 2020 06:41:40
%S A231386 0,0,0,0,11,52,233,1344,8197,49846,351946,2799536,22764021,200196218,
%T A231386 1947350444,19753229932,210793513246,2425636703848,29307938173409,
%U A231386 369141523106550,4920501544208343,68771635812423192,998694091849893095,15169308298544690802
%N A231386 Number of permutations of [n] with exactly one occurrence of one of the consecutive step patterns UUD, UDU, DUU (U=up, D=down).
%H A231386 Alois P. Heinz, <a href="/A231386/b231386.txt">Table of n, a(n) for n = 0..185</a>
%H A231386 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 A231386 S. Kitaev and T. Mansour, <a href="http://www.ru.is/kennarar/sergey/index_files/Papers/multi_avoid_gen_patterns.pdf">On multi-avoidance of generalized patterns</a>
%F A231386 a(n) ~ c * d^n * n! * n, where d = 0.63140578989563018836..., c = 1.015673... . - _Vaclav Kotesovec_, Aug 28 2014
%e A231386 a(4) = 11: 1243, 1342, 2341 (=UUD), 1324, 1423, 2314, 2413, 3412 (=UDU), 2134, 3124, 4123 (=DUU).
%e A231386 a(5) = 52: 12354, 12453, 12543, ..., 53124, 53412, 54123.
%e A231386 a(6) = 233: 123465, 123564, 123654, ..., 653124, 653412, 654123.
%e A231386 a(7) = 1344: 1234576, 1234675, 1234765, ..., 7653124, 7653412, 7654123.
%p A231386 b:= proc(u, o, t) option remember; `if`(t=13, 0, `if`(u+o=0,
%p A231386       `if`(t>6, 1, 0), add(b(u+j-1, o-j,
%p A231386           [2, 3, 3, 6, 12, 9, 8, 9, 9, 12, 13, 13][t]), j=1..o)+
%p A231386       add(b(u-j, o+j-1,
%p A231386           [4, 5, 11, 4, 4, 5, 10, 11, 13, 10, 10, 11][t]), j=1..u)))
%p A231386     end:
%p A231386 a:= n-> add(b(j-1, n-j, 1), j=1..n):
%p A231386 seq(a(n), n=0..30);
%t A231386 b[u_, o_, t_] := b[u, o, t] = If[t==13, 0, If[u + o == 0, If[t > 6, 1, 0],
%t A231386   Sum[b[u+j-1, o-j,
%t A231386     {2, 3, 3, 6, 12, 9, 8, 9, 9, 12, 13, 13}[[t]]], {j, 1, o}] +
%t A231386   Sum[b[u-j, o+j-1,
%t A231386     {4, 5, 11, 4, 4, 5, 10, 11, 13, 10, 10, 11}[[t]]], {j, 1, u}]]];
%t A231386 a[n_] := Sum[b[j - 1, n - j, 1], {j, 1, n}];
%t A231386 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 22 2020, after _Alois P. Heinz_ *)
%Y A231386 Column k=1 of A231384.
%K A231386 nonn
%O A231386 0,5
%A A231386 _Alois P. Heinz_, Nov 08 2013