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A250286 Number of permutations p of [n] such that p(i) > p(i+1) iff i == 0 (mod 9).

%I A250286 #11 Sep 03 2021 01:59:11
%S A250286 1,1,1,1,1,1,1,1,1,1,9,54,219,714,2001,5004,11439,24309,48619,831384,
%T A250286 9069651,64369341,355150566,1635163542,6542615421,23369110326,
%U A250286 75953123676,227864057851,5742168041637,90830731860000,920922875075934,7159714782188364
%N A250286 Number of permutations p of [n] such that p(i) > p(i+1) iff i == 0 (mod 9).
%H A250286 Alois P. Heinz, <a href="/A250286/b250286.txt">Table of n, a(n) for n = 0..500</a>
%p A250286 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A250286      `if`(t=0, add(b(u-j, o+j-1, irem(t+1, 9)), j=1..u),
%p A250286                add(b(u+j-1, o-j, irem(t+1, 9)), j=1..o)))
%p A250286     end:
%p A250286 a:= n-> b(n, 0$2):
%p A250286 seq(a(n), n=0..35);
%t A250286 nmax = 30; CoefficientList[Series[1 + Sum[(x^(9 - k) * HypergeometricPFQ[{1}, {10/9 - k/9, 11/9 - k/9, 4/3 - k/9, 13/9 - k/9, 14/9 - k/9, 5/3 - k/9, 16/9 - k/9, 17/9 - k/9, 2 - k/9}, -x^9/387420489])/(9 - k)!, {k, 0, 8}] / HypergeometricPFQ[{}, {1/9, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 8/9}, -x^9/387420489], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Apr 21 2021 *)
%Y A250286 Row n=9 of A181937.
%K A250286 nonn
%O A250286 0,11
%A A250286 _Alois P. Heinz_, Nov 16 2014