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A261746 Chocolate numbers. 3
1, 2, 4, 6, 24, 56, 120, 720, 1712, 5040, 9408, 40320, 92800, 362880, 3628800, 4948992, 7918592, 39916800, 479001600, 984237056, 6085088256, 6227020800, 63352393728, 87178291200, 168662855680, 1307674368000, 14782316470272, 20922789888000, 38238313152512 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Given an m X n chocolate bar, let A(m,n) be the number of ways to break it into m*n unit pieces where each break occurs along a gridline. Order matters, and the pieces are distinguishable. Then this sequence lists the values A(m,n) in increasing order as m and n range over the positive integers.

The sequence of factorials, A000142, is a subsequence as A(1,n) = A(n,1) = (n-1)!.

For m,n>1, A(m,n) is divisible by 2^(m+n-2).

LINKS

Table of n, a(n) for n=1..29.

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(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 = m = 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(2,2) = 4, and 4 belongs to the sequence.

MATHEMATICA

terms = 29;

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}];

Table[A[m, n], {m, 1, terms}, {n, 1, terms}] // Flatten // Union //

Take[#, terms]&

CROSSREFS

Cf. A257281, A261747, A000142, A261964.

Sequence in context: A261832 A141526 A226169 * A319575 A318609 A195333

Adjacent sequences:  A261743 A261744 A261745 * A261747 A261748 A261749

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified September 24 17:09 EDT 2020. Contains 337321 sequences. (Running on oeis4.)