cp's OEIS Frontend

A334218 Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.

%I A334218 #34 May 17 2020 13:50:01
%S A334218 1,1,0,0,2,0,0,3,3,0,0,4,16,4,0,0,5,55,55,5,0,0,6,156,396,156,6,0,0,7,
%T A334218 399,2114,2114,399,7,0,0,8,960,9528,19328,9528,960,8,0,0,9,2223,38637,
%U A334218 140571,140571,38637,2223,9,0,0,10,5020,146080,882340,1561900,882340,146080,5020,10,0
%N A334218 Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.
%H A334218 Andrew Howroyd, <a href="/A334218/b334218.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F A334218 T(n, k) = n*A008292(n-1, k) for n > 1.
%F A334218 T(n, k) = T(n, n-k) for n > 1.
%F A334218 T(n, k) = n*Sum_{j=0..k} (-1)^j * (k-j)^(n-1) * binomial(n, j) for n > 0.
%e A334218 Triangle begins:
%e A334218   1;
%e A334218   1, 0;
%e A334218   0, 2,   0;
%e A334218   0, 3,   3,    0;
%e A334218   0, 4,  16,    4,     0;
%e A334218   0, 5,  55,   55,     5,    0;
%e A334218   0, 6, 156,  396,   156,    6,   0;
%e A334218   0, 7, 399, 2114,  2114,  399,   7, 0;
%e A334218   0, 8, 960, 9528, 19328, 9528, 960, 8, 0;
%e A334218   ...
%o A334218 (PARI) T(n, k) = {if(n==0, k==0, n*sum(j=0, k, (-1)^j * (k-j)^(n-1) * binomial(n, j)))}
%Y A334218 Columns k=2..9 are A027540(n-1), A151576, A151577, A151578, A151579, A151580, A151581, A151582.
%Y A334218 Row sums are A000142.
%Y A334218 Cf. A008292.
%K A334218 nonn,tabl
%O A334218 0,5
%A A334218 _Andrew Howroyd_, May 04 2020