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A322543 Number of triadic partitions of the unit square into (2n+1) subrectangles. 2
1, 2, 12, 96, 879, 8712, 90972, 985728, 10979577, 124937892, 1446119664, 16972881120, 201526230555, 2416309004872, 29215072931136, 355800894005760, 4360705642282569, 53744080256387478, 665667989498682936, 8281518339078928800, 103441301833577854041, 1296713265300164761632 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A kind of two-dimensional ternary Catalan number. This sequence enumerates the number of decompositions of the unit square into 2n+1 rectangles obtained by the following algorithm.

(a) Start with the unit square.

(b) Perform the following operation n times:

    (1) Choose a rectangle in the current decomposition.

    (2) Trisect this rectangle into three rectangles horizontally or vertically.

Note that different sequences of trisections can produce the same decomposition.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..889

Yu Hin (Gary) Au, Fatemeh Bagherzadeh, Murray R. Bremner, Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube, arXiv:1903.00813 [math.CO], Mar 03 2019.

FORMULA

Recurrence relation: a(n) = C(2n+1) with  C(1) = 1 and C(n) = 2 Sum_{i1,i2,i3} C(i1)C(i2)C(i3) - Sum_{i1,i2,i3,i4,i5,i6,i7,i8,i9} C(i1)C(i2)C(i3)C(i4)C(i5)C(i6)C(i7)C(i8)C(i9). The first sum is over all 3-compositions of n into positive integers (i1+i2+i3=n), and the second sum is over all 9-compositions of n into positive integers (i1+i2+...+i9=n).

a(n) = [x^(2n+1)] G(x), where G(x) satisfies: G(x)^9 - 2*G(x)^3 + G(x) - x = 0.

MAPLE

a:= n-> coeff(series(RootOf(G^9-2*G^3+G-x, G), x, 2*n+2), x, 2*n+1):

seq(a(n), n=0..25);  # Alois P. Heinz, Dec 14 2018

MATHEMATICA

a[n_] := SeriesCoefficient[InverseSeries[x - 2 x^3 + x^9 + O[x]^(2n+2), x], {x, 0, 2n+1}];

Table[a[n], {n, 0, 21}] (* Jean-Fran├žois Alcover, Aug 13 2019, from PARI *)

PROG

(PARI) a(n)={polcoef(serreverse(x - 2*x^3 + x^9 + O(x^(2*n+2))), 2*n+1)} \\ Andrew Howroyd, Dec 14 2018

CROSSREFS

Cf. A000108 (Catalan numbers), A005408, A236339 (decompositions of unit square using bisections).

Sequence in context: A306258 A052691 A292419 * A213422 A307103 A153231

Adjacent sequences:  A322540 A322541 A322542 * A322544 A322545 A322546

KEYWORD

nonn

AUTHOR

Yu Hin Au, Dec 14 2018

STATUS

approved

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Last modified October 15 00:14 EDT 2019. Contains 328025 sequences. (Running on oeis4.)