A370530 - OEIS

%I #23 May 02 2024 04:28:47

%S 0,0,0,0,0,0,0,0,0,1,4,20,116,820,6600,59650,598140,6592740,79232140,

%T 1031214100,14450182880,216911207555,3472641749080,59063810100120,

%U 1063582404217144,20215010963623896,404418346892678160,8494912449554675844,186928503912130116360

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

%H G. C. Greubel, <a href="/A370530/b370530.txt">Table of n, a(n) for n = 0..450</a>

%H 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 G.f.: (1/6) * Sum_{k>=3} k! * x^(k+6) / (1+x^3)^(k+1).

%F a(n) = (1/6) * Sum_{k=0..floor(n/3)-3} (-1)^k * (n-2*k-6)! / k!.

%t 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 (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 (PARI) a(n, k=3, q=3) = sum(j=0, n\q-k, (-1)^j*(n-(q-1)*(j+k))!/j!)/k!;

%o (Magma)

%o [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 (SageMath)

%o [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 Column k=3 of A177250.

%Y Cf. A177251, A370525, A370528.

%K nonn

%O 0,11

%A _Seiichi Manyama_, Feb 21 2024