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

Author

Wouter Meeussen, Aug 03 2001

Keywords

• nonn

Examples

			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)
		

Links

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

Crossrefs

Cf. A047993, A063074, A063746.

Programs

Mathematica

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

PARI

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

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