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A177260
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Number of derangements of {1,2,...,n} having no adjacent 4-cycles (an adjacent 4-cycle is a cycle of the form (i,i+1,i+2,i+3)).
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4
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1, 0, 1, 2, 8, 44, 262, 1846, 14789, 133232, 1333112, 14669758, 176081478, 2289458896, 32056423888, 480890367598, 7694774125983, 130818028518432, 2354820682603399, 44743035640567412, 894883797133726171, 18792952193893804872, 413452012727711517437
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OFFSET
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0,4
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LINKS
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FORMULA
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a(n) = Sum_{s=0..n} Sum_{t=0..floor((n-s)/4)} (-1)^(s+t)*(n-3*t)!/(s!*t!).
Conjecture: D-finite with recurrence a(n) = (n-1)*a(n-1) + (n-1)*a(n-2) + 2*a(n-4) + (n-1)*a(n-5) + 3*a(n-8). - R. J. Mathar, Jul 26 2022
G.f.: Sum_{k>=0} k! * x^k / (1+x+x^4)^(k+1). - Seiichi Manyama, Feb 22 2024
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EXAMPLE
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a(6)=262 because among the 265 (= A000166(6)) derangements of {1,2,3,4,5,6} only (1234)(56), (16)(2345), and (12)(3456) have adjacent 4-cycles.
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MAPLE
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a := proc (n) local ct, t, s: ct := 0: for s from 0 to n do for t from 0 to (1/4)*n do if s+4*t <= n then ct := ct+(-1)^(s+t)*factorial(n-3*t)/(factorial(s)*factorial(t)) else end if end do end do: ct end proc; seq(a(n), n = 0 .. 22);
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MATHEMATICA
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a[n_] := Module[{ct = 0, t, s}, For[s = 0, s <= n, s++, For[t = 0, t <= n/3, t++, If[s + 4*t <= n, ct = ct + (-1)^(s + t)*Factorial[n - 3*t] / (Factorial[s]*Factorial[t])]]]; ct];
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PROG
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(Magma)
F:=Factorial;
A177258:= func< n | (&+[(&+[(-1)^(j+k)*F(n-3*k)/(F(j)*F(k)): k in [0..Floor((n-j)/4)]]): j in [0..n]]) >;
(SageMath)
f=factorial;
def A177260(n): return sum(sum((-1)^(j+k)*f(n-3*k)/(f(j)*f(k)) for k in range(1+(n-j)//4)) for j in range(n+1))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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