A063075 -id:A063075 - cp's OEIS frontend

A063074 Number of partitions of 2n^2 whose Ferrers-plot fits within a 2n X 2n 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.

1, 2, 8, 58, 526, 5448, 61108, 723354, 8908546, 113093022, 1470597342, 19499227828, 262754984020, 3589093760726, 49596793134484, 692260288169282, 9747120868919060, 138298900243896166, 1975688102624819336, 28396056820503468894, 410363630540693436398

Offset: 0

Wouter Meeussen, Aug 03 2001

nonn

Also the number of subsets of {1,...,4*n} containing exactly 2*n elements with total sum n*(4*n+1) (see also A060468 for a related sequence). This is of course the same as the number of partitions of n*(4*n+1) having 2*n distinct parts of length at most 4*n. This number is the coefficient of t^0 q^0 in Product_{k=1..4*n} (t*q^k + 1/(t*q^k)). - Roland Bacher, May 02 2002
A bijection with a dissection as above of the 2n X 2n checkerboard is given by subtracting 1,2,3,...,2n of the smallest, second-smallest, etc. element in the subset. Example for n=2: {1,2,7,8} (yields the checkerboard partition {1-1,2-2,7-3,8-4}={0,0,4,4}), {1,3,6,8} (yields {1-1,3-2,6-3,8-4}={0,1,3,4}), {1,4,5,8} (yields {0,2,2,4}), {1,4,6,7} (yields {0,2,3,3}), {3,4,5,6} (yields {2,2,2,2}), {2,4,5,7} (yields {1,2,2,3}), {2,3,6,7} (yields {1,1,3,3}), {2,3,5,8} (yields {1,1,2,4}).
Appears to be the number of random walks of length 4n, moves +/-1, starting and ending at 0 and with signed area 0 under the path. It would be nice to have a lower bound of the form a(n) > c*2^{4n}/n^d. - David_Mumford(AT)brown.edu, Jun 25 2003

			For a 4 X 4 board (n=2) the 8 partitions are (4,4,0,0), (4,3,1,0), (4,2,1,1), (4,2,2,0), (3,3,2,0), (3,3,1,1), (3,2,2,1), (2,2,2,2).

Alois P. Heinz, Table of n, a(n) for n = 0..100
Helmut Prodinger, On the number of partitions of 1...n into two sets of equal cardinalities and equal sums, Canad. Math. Bull. 25 (1982), pp. 238-241.

Cf. A047993, A063075.

Bisection of row n=2 of A204459. - Alois P. Heinz, Jan 18 2012

b:= proc(n, i, t) option remember;
      `if`(it*(2*i-t+1)/2, 0,
      `if`(n=0, 1, b(n, i-1, t) +`if`(n b(n*(4*n+1), 4*n, 2*n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 18 2012

Table[ Length@Select[ IntegerPartitions[ 2n^2 ], Length[ # ] <= 2n && First[ # ] <= 2n& ], {n, 1, 5} ] or faster: Table[ T[ 2n^2, 2n, 2n ], {n, 0, 24} ] with T[ m, a, b ] as defined in A047993.
(* second program: *)
b[n_, i_, t_] := b[n, i, t] =  If[i < t || n < t (t + 1)/2 || n > t (2i - t + 1)/2, 0, If[n == 0, 1, b[n, i - 1, t] + If[n < i, 0, b[n - i, i - 1, t - 1]]]]; a[n_] := b[n (4n + 1), 4n, 2n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

a(n) = A029895(2n) = A067059(2n, 2n) = A107110(2n, 2n^2). a(n) seems to be close to (sqrt(75)/Pi)*16^n/(20n^2+6n+1). - Henry Bottomley, May 12 2005

More terms from Alois P. Heinz, Jan 18 2012

A125806 Triangle of the sum of squared coefficients of q in the q-binomial coefficients, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 16, 16, 5, 1, 1, 6, 29, 48, 29, 6, 1, 1, 7, 47, 119, 119, 47, 7, 1, 1, 8, 72, 256, 390, 256, 72, 8, 1, 1, 9, 104, 500, 1070, 1070, 500, 104, 9, 1, 1, 10, 145, 900, 2592, 3656, 2592, 900, 145, 10, 1, 1, 11, 195, 1525, 5674, 10762, 10762, 5674, 1525, 195, 11, 1

Offset: 0

Paul D. Hanna, Dec 11 2006

Central terms equal A063075 (number of partitions of 2n^2 whose Ferrers-plot fits within a 2n X 2n box and cover an n X n box).

			The triangle of q-binomial coefficients:
C_q(n,k) = [Product_{i=n-k+1..n}(1-q^i)]/[Product_{j=1..k}(1-q^j)]
begins:
1;
1, 1;
1, 1+q, 1;
1, 1+q+q^2, 1+q+q^2, 1;
1, 1+q+q^2+q^3, 1+q+2*q^2+q^3+q^4, 1+q+q^2+q^3, 1; ...
recurrence: C_q(n+1,k) = C_q(n,k-1) + q^k * C_q(n,k).
Element T(n,k) of this triangle equals the sum of the squares
of the coefficients of q in q-binomial coefficient C_q(n,k).
This triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 8, 4, 1;
1, 5, 16, 16, 5, 1;
1, 6, 29, 48, 29, 6, 1;
1, 7, 47, 119, 119, 47, 7, 1;
1, 8, 72, 256, 390, 256, 72, 8, 1;
1, 9, 104, 500, 1070, 1070, 500, 104, 9, 1;
1, 10, 145, 900, 2592, 3656, 2592, 900, 145, 10, 1;
1, 11, 195, 1525, 5674, 10762, 10762, 5674, 1525, 195, 11, 1;
1, 12, 256, 2456, 11483, 28160, 37834, 28160, 11483, 2456, 256, 12, 1;
The central terms equal A063075:
1, 2, 8, 48, 390, 3656, 37834, 417540, 4836452, 58130756, ...
MATRIX INVERSE.
The matrix inverse starts
1;
-1,1;
1,-2,1;
-1,3,-3,1;
-1,0,4,-4,1;
9,-21,12,4,-5,1;
-1,34,-73,44,1,-6,1;
-219,479,-219,-139,109,-5,-7,1;
101,-1536,3072,-1776,-54,216,-16,-8,1; - _R. J. Mathar_, Mar 22 2013

Paul D. Hanna, Rows n = 0..45, listed in flattened form as n, a(n), for n = 0..1080.

Cf. A063075 (central terms); A125807, A125808, A125809 (row sums).

C := proc(q,n,k)
    local i,j ;
    mul(1-q^i,i=n-k+1..n)/mul(1-q^j,j=1..k) ;
    expand(factor(%)) ;
end proc:
A125806 := proc(n,k)
    local qbin ,q;
    qbin := [coeffs(C(q,n,k),q)] ;
    add( e^2,e=qbin) ;
end proc: # R. J. Mathar, Mar 22 2013

T[n_, k_] := Module[{cc, q}, cc = CoefficientList[QBinomial[n, k, q] // FunctionExpand, q]; cc.cc];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 08 2023 *)

T(n,k)=local(C_q=if(n==0 || k==0,1,prod(j=n-k+1,n,1-q^j)/prod(j=1,k,1-q^j))); sum(i=0,(n-k)*k,polcoeff(C_q,i)^2)
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

A125807 Central terms of odd-indexed rows of triangle A125806: a(n) = A125806(2n+1,n).

Original entry on oeis.org

1, 3, 16, 119, 1070, 10762, 116546, 1330923, 15823388, 194168612, 2444224858, 31422225930, 411141476444, 5460849893348, 73474839110524, 999764999592077, 13738614091375204, 190450074950481408, 2660727794475865450

Offset: 0

Paul D. Hanna, Dec 12 2006

nonn

Central terms of even-indexed rows of triangle A125806 equal A063075 (number of partitions of 2n^2 whose Ferrers-plot fits within a 2n X 2n box and cover an n X n box).

Cf. A125806 (triangle); A063075; A125808, A125809 (row sums).

{a(n)=local(C_q=if(n==0,1,prod(j=n+2,2*n+1,1-q^j)/prod(j=1,n,1-q^j))); sum(i=0,(n+1)*n,polcoeff(C_q,i)^2)}

A125808 Adjacent-to-central terms of even-indexed rows of triangle A125806: a(n) = A125806(2n+2,n).

Original entry on oeis.org

1, 4, 29, 256, 2592, 28160, 322873, 3850352, 47369432, 597565304, 7695966346, 100852014156, 1341310032320, 18067954497864, 246098396499471, 3384883529933828, 46960152641672616, 656538880287562528

Offset: 0

Paul D. Hanna, Dec 12 2006

nonn

Central terms of even-indexed rows of triangle A125806 equal A063075 (number of partitions of 2n^2 whose Ferrers-plot fits within a 2n X 2n box and cover an n X n box).

Cf. A125806 (triangle); A063075; A125807, A125809 (row sums).

{a(n)=local(C_q=if(n==0,1,prod(j=n+3,2*n+2,1-q^j)/prod(j=1,n,1-q^j))); sum(i=0,(n+2)*n,polcoeff(C_q,i)^2)}

A125809 Row sums of triangle A125806.

Original entry on oeis.org

1, 2, 4, 8, 18, 44, 120, 348, 1064, 3368, 10952, 36336, 122570, 419104, 1449672, 5064240, 17844558, 63356072, 226459120, 814323856, 2944055592, 10695723368, 39029679176, 142998497292, 525862368660, 1940381764088, 7182278240848

Offset: 0

Paul D. Hanna, Dec 12 2006

nonn

Triangle A125806 gives the sum of squared coefficients of q in the corresponding q-binomial coefficients.

Cf. A125806 (triangle); A063075, A125807, A125808.

{a(n)=sum(k=0,n,sum(i=0,(n-k)*k, polcoeff(if(n==0,1,prod(j=n-k+1,n,1-q^j)/prod(j=1,k,1-q^j)),i)^2))}

cp's OEIS Frontend

A063074 Number of partitions of 2n^2 whose Ferrers-plot fits within a 2n X 2n 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.

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A125806 Triangle of the sum of squared coefficients of q in the q-binomial coefficients, read by rows.

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A125807 Central terms of odd-indexed rows of triangle A125806: a(n) = A125806(2n+1,n).

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A125808 Adjacent-to-central terms of even-indexed rows of triangle A125806: a(n) = A125806(2n+2,n).

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A125809 Row sums of triangle A125806.

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