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A264461 Number of permutations of [n] with exactly two (possibly overlapping) occurrences of the generalized pattern 23-1.

%I A264461 #10 Sep 28 2020 07:54:12
%S A264461 3,23,152,984,6460,43626,304939,2211467,16649780,130097338,1054226016,
%T A264461 8850736900,76901730751,690749091147,6406953787268,61300205459232,
%U A264461 604367205789092,6133919028981542,64027105979768111,686736004045762143,7562191796264603160
%N A264461 Number of permutations of [n] with exactly two (possibly overlapping) occurrences of the generalized pattern 23-1.
%H A264461 Alois P. Heinz, <a href="/A264461/b264461.txt">Table of n, a(n) for n = 4..500</a>
%e A264461 a(4) = 3: 2341, 3412, 3421.
%e A264461 a(5) = 23: 13452, 14523, 14532, 23415, 23514, 23541, 24351, 25341, 32451, 34125, 34152, 34215, 35124, 35142, 35214, 35412, 35421, 42351, 43512, 43521, 52341, 53412, 53421.
%p A264461 b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
%p A264461       b(u-j, o+j-1), j=1..u) +add(convert(series(
%p A264461       b(u+j-1, o-j)*x^u, x, 3), polynom), j=1..o))
%p A264461     end:
%p A264461 a:= n-> coeff(b(n, 0), x, 2):
%p A264461 seq(a(n), n=4..25);
%t A264461 b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[u-j, o+j-1], {j, 1, u}] + Sum[Series[b[u+j-1, o-j] x^u, {x, 0, 3}] // Normal, {j, 1, o}]];
%t A264461 a[n_] := Coefficient[b[n, 0], x, 2];
%t A264461 a /@ Range[4, 25] (* _Jean-François Alcover_, Sep 28 2020, after Maple *)
%Y A264461 Column k=2 of A260670.
%K A264461 nonn
%O A264461 4,1
%A A264461 _Alois P. Heinz_, Nov 14 2015