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Search: id:A063075
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| A063075 |
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Number of partitions of 2n^2 whose Ferrers-plot fits within a 2n X 2n box and cover an n X n box; number of ways to cut a 2n X 2n chessboard into two equal-area pieces along a non-descending line from lower left to upper right and passing through the center. |
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+0
6
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| 1, 2, 8, 48, 390, 3656, 37834, 417540, 4836452, 58130756, 719541996, 9121965276, 117959864244, 1551101290792, 20689450250926, 279395018584860, 3813887739881184, 52557835511244660, 730403326965323706
(list; graph; listen)
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OFFSET
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0,2
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LINKS
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Paul D. Hanna, Table of n, a(n) for n = 0..70
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FORMULA
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a(n) = Sum_{k=0..n^2} A063746(n,k)^2 ; i.e. equals the sums of the squares of the coefficients of q in the central q-binomial coefficients. - Paul D. Hanna, Dec 12 2006
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EXAMPLE
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For a 6 X 6 board (n=3) the partition (6,6,2,2,2,0) represents a Ferrers plot that does not pass through the center of a 6*6 box.
Central q-binomial coefficients begin:
1;
1 + q;
1 + q + 2*q^2 + q^3 + q^4;
1 + q + 2*q^2 + 3*q^3 + 3*q^4 + 3*q^5 + 3*q^6 + 2*q^7 + q^8 + q^9;
the coefficients of q in these polynomials form the rows of triangle A063746.
The sums of squared terms in rows of A063746 equal this sequence.
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MATHEMATICA
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Table[(#.#)&@Table[T[k, n, n], {k, 0, n^2}], {n, 0, 24}] with T[m, a, b] as defined in A047993.
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CROSSREFS
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Cf. A047993, A063074, A063746.
Adjacent sequences: A063072 A063073 A063074 this_sequence A063076 A063077 A063078
Sequence in context: A000165 A109664 A009812 this_sequence A112541 A052667 A006925
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KEYWORD
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nonn
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AUTHOR
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Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 03 2001
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EXTENSIONS
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Additional comments from Paul D. Hanna (pauldhanna(AT)juno.com), Dec 12 2006
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