A257281 Chocolate square numbers.

1, 1, 4, 9408, 63352393728, 3947339798331748515840, 5732998662938820430255187886059028480, 417673987760293241182652126617960927525362518081132298240

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^(2n-2).

Links

Table of n, a(n) for n=0..7.

Caleb Ji, Tanya Khovanova, Robin Park, Angela Song, Chocolate Numbers, arXiv:1509.06093 [math.CO], (21-September-2015).

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..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.

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[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}];
a[n_] := A[n, n];
Table[a[n], {n, 0, 7}] (* Jean-François Alcover, Dec 12 2018 *)

Crossrefs

Cf. A261746, A261747, A261964.

Sequence in context: A072724 A116271 A274551 * A201392 A058417 A352497

Adjacent sequences: A257278 A257279 A257280 * A257282 A257283 A257284

Keyword

nonn

Author

Caleb Ji, Tanya Khovanova, Robin Park, Angela Song, Apr 19 2015

Status

approved