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A334218
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Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.
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9
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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
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OFFSET
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0,5
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
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FORMULA
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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.
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EXAMPLE
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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;
...
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PROG
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(PARI) T(n, k) = {if(n==0, k==0, n*sum(j=0, k, (-1)^j * (k-j)^(n-1) * binomial(n, j)))}
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CROSSREFS
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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 * A278094 A245487 A074734
Adjacent sequences: A334215 A334216 A334217 * A334219 A334220 A334221
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KEYWORD
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nonn,tabl
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AUTHOR
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Andrew Howroyd, May 04 2020
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STATUS
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approved
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