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A230055 Number of permutations of [n] in which the longest increasing run has length 7.

%I A230055 #23 Jul 19 2018 09:50:40
%S A230055 1,14,181,2360,32010,456720,6881160,109546009,1841298059,32629877967,
%T A230055 608572228291,11923667699474,244964063143590,5267496652725480,
%U A230055 118348438201424761,2773714509551524351,67705791536824698266,1718769199589362743761,45314525515737783596251
%N A230055 Number of permutations of [n] in which the longest increasing run has length 7.
%H A230055 Alois P. Heinz, <a href="/A230055/b230055.txt">Table of n, a(n) for n = 7..200</a>
%F A230055 a(n) = A230051(n) - A177553(n).
%F A230055 E.g.f.: 1/Sum_{n>=0} (8*n+1-x)*x^(8*n)/(8*n+1)! - 1/Sum_{n>=0} (7*n+1-x)*x^(7*n)/(7*n+1)!.
%e A230055 a(7) = 1: 1234567.
%e A230055 a(8) = 14: 12345687, 12345786, 12346785, 12356784, 12456783, 13456782, 21345678, 23456781, 31245678, 41235678, 51234678, 61234578, 71234568, 81234567.
%p A230055 b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1,
%p A230055       `if`(t<k-1, add(b(u+j-1, o-j, t+1, k), j=1..o), 0)+
%p A230055       add(b(u-j, o+j-1, 0, k), j=1..u))
%p A230055     end:
%p A230055 a:= n-> b(n, 0, 0, 7)-b(n, 0, 0, 6):
%p A230055 seq(a(n), n=7..30);
%t A230055 b[u_, o_, t_, k_] := b[u, o, t, k] = If[u + o == 0, 1, If[t < k - 1, Sum[b[u + j - 1, o - j, t + 1, k], {j, 1, o}], 0] + Sum[b[u - j, o + j - 1, 0, k], {j, 1, u}]];
%t A230055 a[n_] := b[n, 0, 0, 7] - b[n, 0, 0, 6];
%t A230055 Table[a[n], {n, 7, 30}] (* _Jean-François Alcover_, Jul 19 2018, after _Alois P. Heinz_ *)
%Y A230055 Column k=7 of A008304.
%K A230055 nonn
%O A230055 7,2
%A A230055 _Alois P. Heinz_, Oct 07 2013