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A186765 Number of permutations of {1,2,...,n} having no increasing even cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1) . A cycle is said to be even if it has an even number of entries.

%I A186765 #28 Feb 05 2025 14:45:27
%S A186765 1,1,1,3,14,70,419,2933,23421,210789,2108144,23189584,278279165,
%T A186765 3617629145,50646737049,759701055735,12155215581362,206638664883154,
%U A186765 3719496008830391,70670424167777429,1413408484443295197,29681578173309199137,652994719769134284068
%N A186765 Number of permutations of {1,2,...,n} having no increasing even cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)<b(2)<b(3)<... .  A cycle is said to be even if it has an even number of entries.
%H A186765 Vincenzo Librandi, <a href="/A186765/b186765.txt">Table of n, a(n) for n = 0..200</a>
%F A186765 a(n) = A186764(n,0).
%F A186765 E.g.f.: exp(1-cosh(z))/(1-z).
%F A186765 a(n) = ((sum(m=1..n,sum(k=1..m,((-1)^k*sum(j=0..k,((sum(i=0..j,(j-2*i)^m*binomial(j, i)))*(-1)^(j-k)*binomial(k, j))/2^j))/k!)/m!))+1)*n!. [_Vladimir Kruchinin_, Apr 25 2011]
%F A186765 a(n) ~ n! * exp(1-cosh(1)). - _Vaclav Kotesovec_, Feb 24 2014
%e A186765 a(3)=3 because we have (1)(2)(3), (132), and (123).
%p A186765 g := exp(1-cosh(z))/(1-z); gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);
%p A186765 # second Maple program:
%p A186765 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
%p A186765       binomial(n-1, j-1)*((j-1)!+irem(j, 2)-1), j=1..n))
%p A186765     end:
%p A186765 seq(a(n), n=0..22);  # _Alois P. Heinz_, Feb 05 2025
%t A186765 CoefficientList[Series[E^(1-Cosh[x])/(1-x), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Feb 24 2014 *)
%o A186765 (Maxima) a(n):=((sum(sum(((-1)^k*sum(((sum((j-2*i)^m*binomial(j,i),i,0,j))*(-1)^(j-k)*binomial(k,j))/2^j,j,0,k))/k!,k,1,m)/m!,m,1,n))+1)*n!; /* _Vladimir Kruchinin_, Apr 25 2011 */
%o A186765 (PARI) my(x='x+O('x^66)); Vec(serlaplace(exp(1-cosh(x))/(1-x))) /* _Joerg Arndt_, Apr 26 2011 */
%Y A186765 Column k=0 of A186764.
%Y A186765 Cf. A186761, A186762, A186766, A186767, A186769, A186770
%K A186765 nonn
%O A186765 0,4
%A A186765 _Emeric Deutsch_, Feb 27 2011