A395602 - OEIS

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.

%I #24 Apr 30 2026 09:39:18

%S 1,1,1,2,4,2,6,29,30,5,20,229,604,250,14,70,1847,12168,13740,2236,42,

%T 252,14974,238848,699310,332842,20979,132,924,121430,4569624,33138675,

%U 42660740,8419334,203748,429,3432,983476,85553528,1484701075,4872907670,2711857491,219829560,2031054,1430

%N 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.

%H Andrei Asinowski, Christian Krattenthaler, Toufik Mansour, <a href="https://arxiv.org/abs/1604.02870">Counting triangulations of some classes of subdivided convex polygons</a>, arXiv:1604.02870 [math.CO], 2016.

%F 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).

%e n\k | 1 2 3 4 5 6

%e ----+-------------------------------------------------------------------

%e 2 | 1, 1, 2, 6, 20, 70, ...

%e 3 | 1, 4, 29, 229, 1847, 14974, ...

%e 4 | 2, 30, 604, 12168, 238848, 4569624, ...

%e 5 | 5, 250, 13740, 699310, 33138675, 1484701075, ...

%e 6 | 14, 2236, 332842, 42660740, 4872907670, 510909185422, ...

%e 7 | 42, 20979, 8419334, 2711857491, 745727424435, 182814912101920, ...

%o (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)));

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

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

%K nonn,tabl

%O 2,4

%A _Seiichi Manyama_, Apr 30 2026