A177251 Number of permutations of [n] having no adjacent 3-cycles, i.e., no cycles of the form (i, i+1, i+2).
1, 1, 2, 5, 22, 114, 697, 4923, 39612, 357899, 3588836, 39556420, 475392841, 6187284605, 86701097310, 1301467245329, 20835850494474, 354382860600678, 6381494425302865, 121290065781743383, 2426510081356069016, 50969474697328055063, 1121571023472780698152
Offset: 0
Keywords
Examples
a(4)=22 because the only permutations of {1,2,3,4} having adjacent 3-cycles are (123)(4) and (1)(234).
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..449
- R. A. Brualdi and Emeric Deutsch, Adjacent q-cycles in permutations, arXiv:1005.0781 [math.CO], 2010.
Programs
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Magma
A177251:= func< n | (&+[(-1)^j*Factorial(n-2*j)/Factorial(j): j in [0..Floor(n/3)]]) >; [A177251(n): n in [0..30]]; // G. C. Greubel, Apr 28 2024
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Maple
a := proc (n) options operator, arrow: sum((-1)^j*factorial(n-2*j)/factorial(j), j = 0 .. floor((1/3)*n)) end proc: seq(a(n), n = 0 .. 22);
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Mathematica
a[n_] := Sum[(-1)^j*(n - 2*j)!/j!, {j, 0, n/3}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 17 2017 *) -
SageMath
def A177251(n): return sum((-1)^j*factorial(n-2*j)/factorial(j) for j in range(1+n//3)) [A177251(n) for n in range(31)] # G. C. Greubel, Apr 28 2024
Formula
a(n) = Sum_{j=0..floor(n/3)} (-1)^j*(n-2*j)!/j!.
a(n) = A177250(n,0).
a(n) - n*a(n-1) = 2*a(n-3) + 3*(-1)^(n/3) if 3 | n, otherwise a(n) - n*a(n-1) = 2*a(n-3).
lim_{n -> oo} a(n)/n! = 1.
The o.g.f. g(z) satisfies z^2*(1+z^3)*g'(z) - (1+z^3)(1-z-2z^3)g(z) + 1 - 2z^3 = 0; g(0)=1.
G.f.: hypergeometric2F0([1,1], [], x/(1+x^3))/(1+x^3). - Mark van Hoeij, Nov 08 2011
D-finite with recurrence a(n) = n*a(n-1) + a(n-3) + (n-3)*a(n-4) + 2*a(n-6). - R. J. Mathar, Jul 26 2022
G.f.: Sum_{k>=0} k! * x^k / (1+x^3)^(k+1). - Seiichi Manyama, Feb 20 2024
Extensions
Crossreferences corrected by Emeric Deutsch, May 09 2010