A279443 Number of ways to place 9 points on an n X n board so that no more than 2 points are on a vertical or horizontal straight line.

0, 0, 0, 0, 22650, 4987800, 240023070, 5219088000, 68483325960, 630486309600, 4456523194200, 25647802519680, 125166919041450, 533442526857240, 2029603476250350, 7011735609715200, 22291042191643680, 65914292362262400, 182880685655641440, 479548000781222400

Offset

1,5

Comments

Column 9 of triangle A279445.

Rotations and reflections of placements are counted.

Links

Heinrich Ludwig, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (19,-171,969,-3876,11628, -27132,50388,-75582,92378,-92378,75582,-50388,27132,-11628,3876, -969,171,-19,1).

Formula

a(n) = (n^18 -204*n^16 +1260*n^15 +6846*n^14 -104076*n^13 +394504*n^12 +128520*n^11 -6237075*n^10 +24018372*n^9 -43820196*n^8 +30400020*n^7 +34251148*n^6 -99199296*n^5 +98504496*n^4 -47779200*n^3 +9434880*n^2)/362880; factorized: a(n) = n^2*(n-1)^2*(n-2)^2*(n-3)^2*(n-4)^2*(n^8 +20*n^7 +26*n^6 -820*n^5 -247*n^4 +9704*n^3 -9104*n^2 -14700*n +16380)/9!.

a(n) = SUM(1<=j<=19, C(19,j)*(-1)^(j-1)*a(n-j)).

G.f.: 30*x^5*(755 +151915*x +4970934*x^2 +49653854*x^3 +187307071*x^4 +275138271*x^5 +119386656*x^6 -31251744*x^7 -19595619*x^8 +1706821*x^9 +667466*x^10 -26334*x^11 -2543*x^12 +17*x^13) / (1 -x)^19. - Colin Barker, Dec 24 2016

Prog

(PARI) concat(vector(4), Vec(30*x^5*(755 +151915*x +4970934*x^2 +49653854*x^3 +187307071*x^4 +275138271*x^5 +119386656*x^6 -31251744*x^7 -19595619*x^8 +1706821*x^9 +667466*x^10 -26334*x^11 -2543*x^12 +17*x^13) / (1 -x)^19 + O(x^30))) \\ Colin Barker, Dec 24 2016

Crossrefs

Cf. A279444, A279445, A197458.

Same problem but 2..8 points: A083374, A279437, A279438, A279439, A279440, A279441, A279442.

Sequence in context: A335397 A251789 A105378 * A287377 A179726 A287249

Adjacent sequences: A279440 A279441 A279442 * A279444 A279445 A279446

Keyword

nonn,easy

Author

Heinrich Ludwig, Dec 24 2016

Status

approved