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A250283 Number of permutations p of [n] such that p(i) > p(i+1) iff i == 0 (mod 6).

%I A250283 #27 Aug 01 2023 12:13:59
%S A250283 1,1,1,1,1,1,1,6,27,83,209,461,923,10284,80991,414961,1671853,5699149,
%T A250283 17116009,278723178,3135810159,22493048843,124606826189,574688719793,
%U A250283 2301250545971,49308397822776,721175428306971,6650954153090521,46893517738791361
%N A250283 Number of permutations p of [n] such that p(i) > p(i+1) iff i == 0 (mod 6).
%H A250283 Alois P. Heinz, <a href="/A250283/b250283.txt">Table of n, a(n) for n = 0..500</a>
%H A250283 Ludwig Seidel, <a href="https://babel.hathitrust.org/cgi/pt?id=hvd.32044092897461&amp;view=1up&amp;seq=176">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the <a href="https://www.hathitrust.org/accessibility">HATHI TRUST Digital Library</a>]
%H A250283 Ludwig Seidel, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1877_0157-0187.pdf">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through <a href="https://de.wikipedia.org/wiki/ZOBODAT">ZOBODAT</a>]
%e A250283 a(6) = 1: 123456.
%e A250283 a(7) = 6: 1234576, 1234675, 1235674, 1245673, 1345672, 2345671.
%p A250283 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A250283      `if`(t=0, add(b(u-j, o+j-1, irem(t+1, 6)), j=1..u),
%p A250283                add(b(u+j-1, o-j, irem(t+1, 6)), j=1..o)))
%p A250283     end:
%p A250283 a:= n-> b(n, 0$2):
%p A250283 seq(a(n), n=0..35);
%t A250283 nmax = 30; CoefficientList[Series[1 + Sum[(x^(6 - k) * HypergeometricPFQ[{1}, {7/6 - k/6, 4/3 - k/6, 3/2 - k/6, 5/3 - k/6, 11/6 - k/6, 2 - k/6}, -x^6/46656])/(6 - k)!, {k, 0, 5}] / HypergeometricPFQ[{}, {1/6, 1/3, 1/2, 2/3, 5/6}, -x^6/46656], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Apr 21 2021 *)
%o A250283 (Sage)
%o A250283 # From _Peter Luschny_, Feb 06 2017 (Start)
%o A250283 @cached_function
%o A250283 def b(u, o, t):
%o A250283     if u ==-o: return 1
%o A250283     if t == 0: return sum(b(u-j, o+j-1, (t+1) % 6) for j in (1..u))
%o A250283     return sum(b(u+j-1, o-j, (t+1) % 6) for j in (1..o))
%o A250283 a = lambda n: b(n, 0, 0)
%o A250283 print([a(n) for n in (0..28)]) # after Maple program
%o A250283 # Alternatively:
%o A250283 @cached_function
%o A250283 def A(m, n):
%o A250283     if n == 0: return 1
%o A250283     s = -1 if m.divides(n) else 1
%o A250283     t = [m*k for k in (0..(n-1)//m)]
%o A250283     return s*add(binomial(n, k)*A(m, k) for k in t)
%o A250283 A250283 = lambda n: (-1)^int(is_odd(n//6))*A(6, n)
%o A250283 print([A250283(n) for n in (0..28)])
%o A250283 # (End)
%Y A250283 Row n=6 of A181937.
%K A250283 nonn
%O A250283 0,8
%A A250283 _Alois P. Heinz_, Nov 16 2014