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A334772 Array read by antidiagonals: T(n,k) is the number of permutations of k indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima. 6
2, 12, 66, 36, 576, 1168, 80, 2610, 17376, 16220, 150, 8520, 129800, 448800, 202416, 252, 22680, 659560, 5748750, 10861056, 2395540, 392, 52416, 2596608, 46412200, 241987500, 253940736, 27517568, 576, 109116, 8505728, 273322980, 3121135440, 9885006250, 5807161344, 310123764 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

T(n,k) is divisible by n and 2*T(n,k) is divisible by n*k.

LINKS

Table of n, a(n) for n=2..37.

FORMULA

T(n,k) = n*k*( P(k,4)^(n-2) * P(k-2,2)^2 + 4*(Sum_{j=0..n-3} P(k-1,3) * P(k-2,2) * P(k,2)^j * P(k, 4)^(n-j-3)) + 4*(Sum_{j=0..n-4} (j + 1) * P(k-1,3)^2 * P(k,2)^j * P(k,4)^(n-j-4)) )/2 where P(n,k) = binomial(n+k-1, k-1).

T(n,k) = n*k*( (k^2 + 4*k + 1)^2*binomial(k+3, 3)^(n-2) + 12*(k + 2)*(k+1)^(n-2) - 6*k*(k+5)*n*(k+1)^(n-2))/(2*(k + 5)^2).

EXAMPLE

Array begins:

==========================================================

n\k |        2          3            4              5

----|----------------------------------------------------

  2 |        2         12           36             80 ...

  3 |       66        576         2610           8520 ...

  4 |     1168      17376       129800         659560 ...

  5 |    16220     448800      5748750       46412200 ...

  6 |   202416   10861056    241987500     3121135440 ...

  7 |  2395540  253940736   9885006250   203933233280 ...

  8 | 27517568 5807161344 395426250000 13051880894720 ...

...

The T(2,3) = 12 permutations of 111222 with 2 local maxima are 112122, 112212 and their rotations.

The T(3,2) = 66 permutations of 112233 with 2 local maxima are 112323, 113223, 113232, 121233, 121332, 122133, 122313, 123213, 123123, 123132, 131322 and their rotations.

PROG

(PARI) T(n, k)={n*k*( (k^2 + 4*k + 1)^2*binomial(k+3, 3)^(n-2) + 12*(k + 2)*(k+1)^(n-2) - 6*k*(k+5)*n*(k+1)^(n-2))/(2*(k + 5)^2)}

CROSSREFS

Columns k=2..6 are A159716, A159722, A159728, A159734, A159737.

Sequence in context: A097632 A076804 A190153 * A197745 A323925 A039633

Adjacent sequences:  A334769 A334770 A334771 * A334773 A334774 A334775

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, May 10 2020

STATUS

approved

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Last modified September 29 21:59 EDT 2020. Contains 337432 sequences. (Running on oeis4.)