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A287328 Number of permutations of [n] with exactly n (possibly overlapping) occurrences of the generalized pattern 13-2 (alternatively: 2-13, 2-31, or 31-2).

%I A287328 #15 Dec 06 2023 13:53:11
%S A287328 1,0,0,0,0,0,2,42,716,11115,166144,2440438,35573362,517145226,
%T A287328 7517489564,109427233994,1596263948916,23344423478564,342325226476816,
%U A287328 5033781463313268,74223067319020872,1097343555359525542,16265399526715750734,241690082996256941332
%N A287328 Number of permutations of [n] with exactly n (possibly overlapping) occurrences of the generalized pattern 13-2 (alternatively: 2-13, 2-31, or 31-2).
%C A287328 Also number of permutations of [n] with n nestings; also number of permutations of [n] with n crossings.
%H A287328 Alois P. Heinz, <a href="/A287328/b287328.txt">Table of n, a(n) for n = 0..300</a>
%F A287328 a(n) = A263776(n,n).
%p A287328 b:= proc(u, o) option remember;
%p A287328       `if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
%p A287328        add(expand(b(u+j-1, o-j)*x^(j-1)), j=1..o))
%p A287328     end:
%p A287328 a:= n-> coeff(b(n, 0), x, n):
%p A287328 seq(a(n), n=0..30);
%t A287328 b[u_, o_] := b[u, o] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1], {j, 1, u}] + Sum[Expand[b[u + j - 1, o - j]*x^(j - 1)], {j, 1, o}]];
%t A287328 a[n_] := Coefficient[b[n, 0], x, n];
%t A287328 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Dec 06 2023, after _Alois P. Heinz_ *)
%Y A287328 Main diagonal of A263776.
%K A287328 nonn
%O A287328 0,7
%A A287328 _Alois P. Heinz_, Aug 31 2017