A063075 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.

1, 2, 8, 48, 390, 3656, 37834, 417540, 4836452, 58130756, 719541996, 9121965276, 117959864244, 1551101290792, 20689450250926, 279395018584860, 3813887739881184, 52557835511244660, 730403326965323706

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..450 (terms 0..80 from Paul D. Hanna)

Formula

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

a(n) = [q^(n^2)](Product_{j=1..n} (1-q^(n+j))/(1-q^j))^2. - Tani Akinari, Jan 28 2022

a(n) ~ sqrt(3) * 2^(4*n - 1/2) / (Pi^(3/2) * n^(5/2)). - Vaclav Kotesovec, Feb 02 2022

Example

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.
From Paul D. Hanna, Dec 12 2006: (Start)
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. (End)

Mathematica

Table[(#.#)&@Table[T[k, n, n], {k, 0, n^2}], {n, 0, 24}] (* with T[m, a, b] as defined in A047993 *)

Prog

(PARI) a(n)=polcoef((prod(j=1, n, (1-q^(n+j))/(1-q^j)))^2, n^2, q) \\ Tani Akinari, Jan 28 2022

Crossrefs

Cf. A047993, A063074, A063746.

Sequence in context: A232005 A351422 A229233 * A177386 A112541 A052667

Adjacent sequences: A063072 A063073 A063074 * A063076 A063077 A063078

Keyword

nonn

Author

Wouter Meeussen, Aug 03 2001

Status

approved