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A264781 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 45321; triangle T(n,k), n >= 0, 0 <= k <= max(0, floor((n-1)/4)), read by rows.

%I A264781 #36 May 20 2023 23:17:05
%S A264781 1,1,2,6,24,119,1,708,12,4914,126,38976,1344,347765,15110,5,3447712,
%T A264781 180736,352,37598286,2308548,9966,447294144,31481472,225984,
%U A264781 5764747515,457520055,4753185,45,80011430240,7068885600,97954080,21280,1189835682714,115808906178
%N A264781 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 45321; triangle T(n,k), n >= 0, 0 <= k <= max(0, floor((n-1)/4)), read by rows.
%C A264781 Consecutive patterns 12354, 21345, 54312 give the same triangle.
%C A264781 The attached Maple program gives a recurrence for the o.g.f. of each row in terms of u. Using that recurrence we may get any row or column from this irregular triangular array T(n,k). The recurrence follows from manipulation of the bivariate o.g.f./e.g.f. 1/W(u,z) = Sum_{n, k >= 0} T(n, k)*u^k*z^n/n!, whose reciprocal W(u,z) is the solution of the o.d.e. in Theorem 3.2 in Elizalde and Noy (2003) (with m = a = 3). - _Petros Hadjicostas_, Nov 05 2019
%H A264781 Alois P. Heinz, <a href="/A264781/b264781.txt">Rows n = 0..170, flattened</a>
%H A264781 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>, 2011.
%H A264781 Sergi Elizalde and Marc Noy, <a href="https://doi.org/10.1016/S0196-8858(02)00527-4 ">Consecutive patterns in permutations</a>, Adv. Appl. Math. 30 (2003), 110-125; see Theorem 3.2 (p. 116) with m = a = 3.
%H A264781 Petros Hadjicostas, <a href="/A264781/a264781_1.txt">Maple program for a recurrence</a>.
%F A264781 Sum_{k > 0} k * T(n,k) = A062199(n-5) for n > 4.
%e A264781 T(5,1) = 1: 45321.
%e A264781 T(6,1) = 12: 156432, 256431, 356421, 453216, 456321, 463215, 546321, 563214, 564213, 564312, 564321, 645321.
%e A264781 T(9,2) = 5: 786549321, 796548321, 896547321, 897546321, 897645321.
%e A264781 Triangle T(n,k) begins:
%e A264781 00 :           1;
%e A264781 01 :           1;
%e A264781 02 :           2;
%e A264781 03 :           6;
%e A264781 04 :          24;
%e A264781 05 :         119,          1;
%e A264781 06 :         708,         12;
%e A264781 07 :        4914,        126;
%e A264781 08 :       38976,       1344;
%e A264781 09 :      347765,      15110,        5;
%e A264781 10 :     3447712,     180736,      352;
%e A264781 11 :    37598286,    2308548,     9966;
%e A264781 12 :   447294144,   31481472,   225984;
%e A264781 13 :  5764747515,  457520055,  4753185,    45;
%e A264781 14 : 80011430240, 7068885600, 97954080, 21280;
%p A264781 b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(
%p A264781        b(u+j-1, o-j, `if`(u+j-3<j, 0, j)), j=1..o)+ expand(
%p A264781       `if`(t=-2, x, 1)*add(b(u-j, o+j-1, `if`(j<t or t=-2, 0,
%p A264781       `if`(t>0, -1, `if`(t=-1, -2, 0)))), j=1..u)))
%p A264781     end:
%p A264781 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
%p A264781 seq(T(n), n=0..17);
%t A264781 b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[
%t A264781      b[u+j-1, o-j, If[u+j-3 < j, 0, j]], {j, 1, o}] + Expand[
%t A264781      If[t == -2, x, 1]*Sum[b[u-j, o+j-1, If[j < t || t == -2, 0,
%t A264781      If[t > 0, -1, If[t == -1, -2, 0]]]], {j, 1, u}]]];
%t A264781 T[n_] := CoefficientList[b[n, 0, 0], x];
%t A264781 T /@ Range[0, 17] // Flatten (* _Jean-François Alcover_, Feb 19 2021, after _Alois P. Heinz_ *)
%Y A264781 Columns k=0-1 give: A202213, A264896.
%Y A264781 Row sums give A000142.
%Y A264781 T(4n+1,n) gives A007696.
%Y A264781 Cf. A002265, A007696, A062199.
%K A264781 nonn,tabf
%O A264781 0,3
%A A264781 _Alois P. Heinz_, Nov 24 2015