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A250285 Number of permutations p of [n] such that p(i) > p(i+1) iff i == 0 (mod 8).

%I A250285 #11 Sep 03 2021 01:59:18
%S A250285 1,1,1,1,1,1,1,1,1,8,44,164,494,1286,3002,6434,12869,194464,1925200,
%T A250285 12394480,62224336,261667792,959874928,3154435120,9464040829,
%U A250285 210311057024,3007458113984,27514536974144,193384741516784,1123028832217904,5617639404687824
%N A250285 Number of permutations p of [n] such that p(i) > p(i+1) iff i == 0 (mod 8).
%H A250285 Alois P. Heinz, <a href="/A250285/b250285.txt">Table of n, a(n) for n = 0..500</a>
%p A250285 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A250285      `if`(t=0, add(b(u-j, o+j-1, irem(t+1, 8)), j=1..u),
%p A250285                add(b(u+j-1, o-j, irem(t+1, 8)), j=1..o)))
%p A250285     end:
%p A250285 a:= n-> b(n, 0$2):
%p A250285 seq(a(n), n=0..35);
%t A250285 nmax = 30; CoefficientList[Series[1 + Sum[(x^(8 - k) * HypergeometricPFQ[{1}, {9/8 - k/8, 5/4 - k/8, 11/8 - k/8, 3/2 - k/8, 13/8 - k/8, 7/4 - k/8, 15/8 - k/8, 2 - k/8}, -x^8/16777216])/(8 - k)!, {k, 0, 7}] / HypergeometricPFQ[{}, {1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8}, -x^8/16777216], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Apr 21 2021 *)
%Y A250285 Row n=8 of A181937.
%K A250285 nonn
%O A250285 0,10
%A A250285 _Alois P. Heinz_, Nov 16 2014