cp's OEIS Frontend

A370530 Number of permutations of [n] having exactly three adjacent 3-cycles.

%I A370530 #23 May 02 2024 04:28:47
%S A370530 0,0,0,0,0,0,0,0,0,1,4,20,116,820,6600,59650,598140,6592740,79232140,
%T A370530 1031214100,14450182880,216911207555,3472641749080,59063810100120,
%U A370530 1063582404217144,20215010963623896,404418346892678160,8494912449554675844,186928503912130116360
%N A370530 Number of permutations of [n] having exactly three adjacent 3-cycles.
%H A370530 G. C. Greubel, <a href="/A370530/b370530.txt">Table of n, a(n) for n = 0..450</a>
%H A370530 R. A. Brualdi and Emeric Deutsch, <a href="http://arxiv.org/abs/1005.0781">Adjacent q-cycles in permutations</a>, arXiv:1005.0781 [math.CO], 2010.
%F A370530 G.f.: (1/6) * Sum_{k>=3} k! * x^(k+6) / (1+x^3)^(k+1).
%F A370530 a(n) = (1/6) * Sum_{k=0..floor(n/3)-3} (-1)^k * (n-2*k-6)! / k!.
%t A370530 Table[Sum[(-1)^k*(n-2*k-6)!/k!, {k,0,Floor[(n-9)/3]}]/6, {n,0,30}] (* _G. C. Greubel_, May 01 2024 *)
%o A370530 (PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(k=3, N, k!*x^(k+6)/(1+x^3)^(k+1))/6))
%o A370530 (PARI) a(n, k=3, q=3) = sum(j=0, n\q-k, (-1)^j*(n-(q-1)*(j+k))!/j!)/k!;
%o A370530 (Magma)
%o A370530 [n le 8 select 0 else (&+[(-1)^k*Factorial(n-2*k-6)/Factorial(k): k in [0..Floor((n-9)/3)]])/6: n in [0..30]]; // _G. C. Greubel_, May 01 2024
%o A370530 (SageMath)
%o A370530 [sum((-1)^k*factorial(n-2*k-6)/factorial(k) for k in range(1+(n-9)//3))/6 for n in range(31)] # _G. C. Greubel_, May 01 2024
%Y A370530 Column k=3 of A177250.
%Y A370530 Cf. A177251, A370525, A370528.
%K A370530 nonn
%O A370530 0,11
%A A370530 _Seiichi Manyama_, Feb 21 2024