A395602 Square array A(n,k), n >= 2, k >= 1, read by antidiagonals downwards, where A(n,k) = [x^(n*k-2)] (1-x) * (1+x)^(n*k-4) * (Sum_{j=0..k} x^j)^n.

1, 1, 1, 2, 4, 2, 6, 29, 30, 5, 20, 229, 604, 250, 14, 70, 1847, 12168, 13740, 2236, 42, 252, 14974, 238848, 699310, 332842, 20979, 132, 924, 121430, 4569624, 33138675, 42660740, 8419334, 203748, 429, 3432, 983476, 85553528, 1484701075, 4872907670, 2711857491, 219829560, 2031054, 1430

Offset

2,4

Links

Table of n, a(n) for n=2..46.

Andrei Asinowski, Christian Krattenthaler, Toufik Mansour, Counting triangulations of some classes of subdivided convex polygons, arXiv:1604.02870 [math.CO], 2016.

Formula

A(n,k) = Sum_{i=0..n*k-2} binomial(n*k-4,i) * Sum_{j=0..floor((n*k-i-2)/(k+1))} (-1)^j * binomial(n,j) * binomial(n*(k+1)-i-4-(k+1)*j,n-2).

Example

n\k |  1      2        3           4             5                6
----+-------------------------------------------------------------------
  2 |  1,     1,       2,          6,           20,              70, ...
  3 |  1,     4,      29,        229,         1847,           14974, ...
  4 |  2,    30,     604,      12168,       238848,         4569624, ...
  5 |  5,   250,   13740,     699310,     33138675,      1484701075, ...
  6 | 14,  2236,  332842,   42660740,   4872907670,    510909185422, ...
  7 | 42, 20979, 8419334, 2711857491, 745727424435, 182814912101920, ...

Prog

(PARI) a(n, k) = sum(i=0, n*k-2, binomial(n*k-4, i)*sum(j=0, (n*k-i-2)\(k+1), (-1)^j*binomial(n, j)*binomial(n*(k+1)-i-4-(k+1)*j, n-2)));

Crossrefs

Columns k=2..4 give A086452(n-2), A282735, A282736.

Rows n=2..4 give A087809(k-1), A282733(k-1), A282734(k-1).

Sequence in context: A229460 A154120 A361727 * A261964 A177847 A296471

Adjacent sequences: A395599 A395600 A395601 * A395603 A395605 A395606

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 30 2026

Status

approved