%I #47 Dec 12 2018 03:17:19
%S 1,1,4,9408,63352393728,3947339798331748515840,
%T 5732998662938820430255187886059028480,
%U 417673987760293241182652126617960927525362518081132298240
%N Chocolate square numbers.
%C Given an n X n chocolate bar, a(n) is the number of ways to break it into n^2 unit pieces where each break occurs along a grid line. Order matters, and the pieces are distinguishable.
%C a(n) is divisible by 2^(2n-2).
%H Caleb Ji, Tanya Khovanova, Robin Park, Angela Song, <a href="http://arxiv.org/abs/1509.06093">Chocolate Numbers</a>, arXiv:1509.06093 [math.CO], (21-September-2015).
%H Caleb Ji, Tanya Khovanova, Robin Park, Angela Song, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Khovanova/khova9.html">Chocolate Numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), #16.1.7.
%F a(n) = A(n,n) with A(m,n)=1 for max(m,n)<2 and A(m,n) = Sum_{i=1..m-1} C(m*n-2,i*n-1) *A(i,n) *A(m-i,n) + Sum_{i=1..n-1} C(m*n-2,i*m-1) *A(m,i) *A(m,n-i) else.
%e For n = 2, there are two ways for the first break: breaking it horizontally or vertically. After that we need two more breaks, which can be done in either order. Thus a(2) = 4.
%t A[m_, n_] := A[m, n] = If[Max[m, n]<2, 1, Sum[A[i, n] Binomial[m n - 2, i n - 1] A[m-i, n], {i, 1, m-1}]] + Sum[A[m, i] Binomial[m n - 2, i m - 1] A[m, n-i], {i, 1, n-1}];
%t a[n_] := A[n, n];
%t Table[a[n], {n, 0, 7}] (* _Jean-François Alcover_, Dec 12 2018 *)
%Y Cf. A261746, A261747, A261964.
%K nonn
%O 0,3
%A _Caleb Ji_, _Tanya Khovanova_, _Robin Park_, _Angela Song_, Apr 19 2015