A261747 Chocolate-2 numbers.

1, 1, 4, 56, 1712, 92800, 7918592, 984237056, 168662855680, 38238313152512, 11106033743298560, 4026844843819663360, 1784377436257886142464, 949324216111786046259200, 597340801661667138076672000, 438858704839955952346364641280

Offset

0,3

Comments

Given a 2-by-n chocolate bar, a(n) is the number of ways to break it into 2n unit pieces where each break occurs along a gridline. Order matters, and the pieces are distinguishable.

For n>1, a(n) is divisible by 2^n.

Links

Table of n, a(n) for n=0..15.

Caleb Ji, Tanya Khovanova, Robin Park, Angela Song, Chocolate Numbers, arXiv:1509.06093 [math.CO], 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,2) 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 that can be done in any order. Thus a(2) = 4.

Crossrefs

Cf. A257281, A261746, A261964.

Sequence in context: A113583 A174489 A009563 * A111849 A009159 A013055

Adjacent sequences: A261744 A261745 A261746 * A261748 A261749 A261750

Keyword

nonn

Author

Caleb Ji, Tanya Khovanova, Robin Park, Angela Song, Aug 30 2015

Status

approved