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 A334218 Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents. 9
 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, 399, 2114, 2114, 399, 7, 0, 0, 8, 960, 9528, 19328, 9528, 960, 8, 0, 0, 9, 2223, 38637, 140571, 140571, 38637, 2223, 9, 0, 0, 10, 5020, 146080, 882340, 1561900, 882340, 146080, 5020, 10, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50) FORMULA T(n, k) = n*A008292(n-1, k) for n > 1. T(n, k) = T(n, n-k) for n > 1. T(n, k) = n*Sum_{j=0..k} (-1)^j * (k-j)^(n-1) * binomial(n, j) for n > 0. EXAMPLE Triangle begins: 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, 399, 2114, 2114, 399, 7, 0; 0, 8, 960, 9528, 19328, 9528, 960, 8, 0; ... PROG (PARI) T(n, k) = {if(n==0, k==0, n*sum(j=0, k, (-1)^j * (k-j)^(n-1) * binomial(n, j)))} CROSSREFS Columns k=2..9 are A027540(n-1), A151576, A151577, A151578, A151579, A151580, A151581, A151582. Row sums are A000142. Cf. A008292. Sequence in context: A188122 A341841 A050186 * A342984 A342985 A278094 Adjacent sequences: A334215 A334216 A334217 * A334219 A334220 A334221 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, May 04 2020 STATUS approved

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Last modified December 7 14:20 EST 2022. Contains 358656 sequences. (Running on oeis4.)