

A257281


Chocolate square numbers.


3




OFFSET

0,3


COMMENTS

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.
a(n) is divisible by 2^(2n2).


LINKS

Caleb Ji, Tanya Khovanova, Robin Park, Angela Song, Chocolate Numbers, arXiv:1509.06093 [math.CO], (21September2015).
Caleb Ji, Tanya Khovanova, Robin Park, Angela Song, Chocolate Numbers, Journal of Integer Sequences, Vol. 19 (2016), #16.1.7.


FORMULA

a(n) = A(n,n) with A(m,n)=1 for max(m,n)<2 and A(m,n) = Sum_{i=1..m1} C(m*n2,i*n1) *A(i,n) *A(mi,n) + Sum_{i=1..n1} C(m*n2,i*m1) *A(m,i) *A(m,ni) else.


EXAMPLE

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.


MATHEMATICA

A[m_, n_] := A[m, n] = If[Max[m, n]<2, 1, Sum[A[i, n] Binomial[m n  2, i n  1] A[mi, n], {i, 1, m1}]] + Sum[A[m, i] Binomial[m n  2, i m  1] A[m, ni], {i, 1, n1}];
a[n_] := A[n, n];


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



