cp's OEIS Frontend

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.

%I A063075 #30 Jun 24 2022 14:54:39
%S A063075 1,2,8,48,390,3656,37834,417540,4836452,58130756,719541996,9121965276,
%T A063075 117959864244,1551101290792,20689450250926,279395018584860,
%U A063075 3813887739881184,52557835511244660,730403326965323706
%N 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.
%H A063075 Vaclav Kotesovec, <a href="/A063075/b063075.txt">Table of n, a(n) for n = 0..450</a> (terms 0..80 from Paul D. Hanna)
%F A063075 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
%F A063075 a(n) = [q^(n^2)](Product_{j=1..n} (1-q^(n+j))/(1-q^j))^2. - _Tani Akinari_, Jan 28 2022
%F A063075 a(n) ~ sqrt(3) * 2^(4*n - 1/2) / (Pi^(3/2) * n^(5/2)). - _Vaclav Kotesovec_, Feb 02 2022
%e A063075 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.
%e A063075 From _Paul D. Hanna_, Dec 12 2006: (Start)
%e A063075 Central q-binomial coefficients begin:
%e A063075   1;
%e A063075   1 + q;
%e A063075   1 + q + 2*q^2 + q^3 + q^4;
%e A063075   1 + q + 2*q^2 + 3*q^3 + 3*q^4 + 3*q^5 + 3*q^6 + 2*q^7 + q^8 + q^9;
%e A063075 the coefficients of q in these polynomials form the rows of triangle A063746.
%e A063075 The sums of squared terms in rows of A063746 equal this sequence. (End)
%t A063075 Table[(#.#)&@Table[T[k, n, n], {k, 0, n^2}], {n, 0, 24}] (* with T[m, a, b] as defined in A047993 *)
%o A063075 (PARI) a(n)=polcoef((prod(j=1,n,(1-q^(n+j))/(1-q^j)))^2,n^2,q) \\ _Tani Akinari_, Jan 28 2022
%Y A063075 Cf. A047993, A063074, A063746.
%K A063075 nonn
%O A063075 0,2
%A A063075 _Wouter Meeussen_, Aug 03 2001