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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 * A278094 A245487 A074734

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 February 26 17:22 EST 2021. Contains 341632 sequences. (Running on oeis4.)