This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378495 #68 Nov 30 2024 12:54:12
%S A378495 0,0,0,0,2,0,0,6,9,3,0,24,24,44,20,0,160,225,175,265,145,0,1140,1224,
%T A378495 1434,1350,1854,1134,0,8988,11025,12313,12145,11473,14833,9793,0,
%U A378495 80864,93456,100232,106280,113336,107576,133496,93176,0,809856,965601,1057761,1141425,1108161,1162161,1108161,1334961,972081
%N A378495 Triangle read by rows: T(n,k) is the number of derangements in S_n with no k-cycles. 1 <= k <= n.
%C A378495 A derangement is a permutation with no fixed points.
%C A378495 Conjecture: For n >= 3, the GCD of the n-th row is n-1.
%F A378495 T(n,1) = 0.
%F A378495 T(n,k) = Sum_{i=0..n} (-1)^i*binomial(n,i)*A122974(n-i,k) for k > 1.
%F A378495 T(n,2) = A038205(n).
%F A378495 T(n,n-1) = A000166(n) for n >= 3.
%F A378495 T(n,n) = A000166(n) - (n-1)! for n >= 3.
%F A378495 Conjecture: T(n,n-1) - T(n,n-2) = abs(A238474(n-4)) for n >= 4.
%F A378495 Conjecture: T(n,n-2) - T(n,n) = (n-3)!*(n-4)*(n-1)/2 for n >= 5.
%e A378495 Triangle begins:
%e A378495 | 1 2 3 4 5 6 7 8 9
%e A378495 ---+---------------------------------------------------------------
%e A378495 1 | 0
%e A378495 2 | 0, 0
%e A378495 3 | 0, 2, 0
%e A378495 4 | 0, 6, 9, 3
%e A378495 5 | 0, 24, 24, 44, 20
%e A378495 6 | 0, 160, 225, 175, 265, 145
%e A378495 7 | 0, 1140, 1224, 1434, 1350, 1854, 1134
%e A378495 8 | 0, 8988, 11025, 12313, 12145, 11473, 14833, 9793
%e A378495 9 | 0, 80864, 93456, 100232, 106280, 113336, 107576, 133496, 93176
%Y A378495 Cf. A000166, A038205, A122974, A238474.
%K A378495 nonn,tabl
%O A378495 1,5
%A A378495 _Peter Kagey_, Nov 29 2024