A154604 -id:A154604 - cp's OEIS frontend

A186364 Number of cycle-up-down permutations of {1,2,...,n} having no fixed points. A permutation is said to be cycle-up-down if it is a product of up-down cycles. A cycle (b(1), b(2), ...) is said to be up-down if, when written with its smallest element in the first position, it satisfies b(1)b(3)<... .

1, 0, 1, 1, 5, 15, 71, 341, 1945, 12135, 84091, 635281, 5212085, 46091955, 437198711, 4426839821, 47657861425, 543551916975, 6546911178931, 83039587809961, 1106307936885965, 15445529882517195, 225502102290364751, 3436240674908121701, 54555087491802061705

Offset: 0

Emeric Deutsch, Feb 28 2011

nonn

a(n) = A186363(n,0).

Hankel transform is A154604. Binomial transform is A000111(n+1). - Paul Barry, Apr 11 2011

			a(4) = 5 because we have (12)(34),(13)(24),(1324),(1423), and (14)(23).

Nathaniel Johnston, Table of n, a(n) for n = 0..198
Emeric Deutsch and Sergi Elizalde, Cycle up-down permutations, arXiv:0909.5199 [math.CO], 2009; and also, Australas. J. Combin. 50 (2011), 187-199.

Cf. A186363.

g := exp(-z)/(1-sin(z)): gser := series(g, z = 0, 28): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 24);

CoefficientList[Series[E^(-x)/(1-Sin[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)

E.g.f.: exp(-z)/(1-sin(z)).
G.f.: 1/(1-x^2/(1-x-3*x^2/(1-2*x-6*x^2/(1-3*x-10*x^2/(1-.../(1-n*x-((n+1)*(n+2)/2)*x^2/(1-... (continued fraction). - Paul Barry, Apr 11 2011
a(n) ~ n! * n * exp(-Pi/2) * 2^(n+3) / Pi^(n+2). - Vaclav Kotesovec, Oct 08 2013
G.f.: conjecture: T(0), where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(1-x*k)*(1-x*(k+1))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2013

A154603 Binomial transform of reduced tangent numbers (A002105).

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 110, 400, 1757, 7861, 41402, 220540, 1358183, 8405203, 59340710, 418689544, 3335855897, 26440317193, 234747589106, 2065458479476, 20224631361251, 195625329965671, 2094552876276830, 22092621409440256

Offset: 0

Paul Barry, Jan 12 2009

Hankel transform is A154604.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A002105, A154604.

A002105:= func< n | (-1)^(n+1)*2^n*(4^n - 1)*Bernoulli(2*n)/n >;
b:= func< n | (n mod 2) eq 0 select A002105(Floor(n/2)+1) else 0 >;
A154603:= func< n | (&+[Binomial(n,k)*b(k): k in [0..n]]) >;
[A154603(n): n in [0..30]]; // G. C. Greubel, Sep 20 2024

With[{nn=30},CoefficientList[Series[Exp[x]Sec[x/Sqrt[2]]^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 30 2013 *)

def A002105(n): return (-1)^(n+1)*2^n*(4^n -1)*bernoulli(2*n)/n
def b(n): return A002105(n//2 +1) if n%2==0 else 0
def A154603(n): return sum(binomial(n,k)*b(k) for k in range(n+1))
[A154603(n) for n in range(31)] # G. C. Greubel, Sep 20 2024

G.f: 1/(1-x-x^2/(1-x-3x^2/(1-x-6x^2/(1-x-10x^2/(1-x-15x^2..... (continued fraction);
E.g.f.: exp(x)*(sec(x/sqrt(2))^2);
G.f.: 1/(x*Q(0)), where Q(k)= 1/x - 1 - (k+1)*(k+2)/2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 26 2013
G.f.: 1/Q(0), where Q(k)= 1 - x - 1/2*x^2*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 04 2013
a(n) ~ n! * 2^(2+n/2)*n*(exp(sqrt(2)*Pi)+(-1)^n) / (Pi^(n+2)*exp(Pi/sqrt(2))). - Vaclav Kotesovec, Oct 02 2013
G.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(1-x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 14 2013
a(n) = Sum_{k=0..n} binomial(n,k)*b(k), where b(n) = A002105((n+2)/2) if n mod 2 = 0 otherwise b(n) = 0. - G. C. Greubel, Sep 20 2024

Typo in e.g.f. fixed by Vaclav Kotesovec, Oct 02 2013

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