A063074 -id:A063074 - cp's OEIS frontend

A204459 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of k-element subsets that can be chosen from {1,2,...,kn} having element sum k(k*n+1)/2.

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 2, 1, 1, 0, 1, 0, 3, 0, 1, 0, 1, 8, 8, 4, 1, 1, 0, 1, 0, 33, 0, 5, 0, 1, 0, 1, 58, 141, 86, 25, 6, 1, 1, 0, 1, 0, 676, 0, 177, 0, 7, 0, 1, 0, 1, 526, 3370, 3486, 1394, 318, 50, 8, 1, 1, 0, 1, 0, 17575, 0, 11963, 0, 519, 0, 9, 0, 1

Offset: 0

Alois P. Heinz, Jan 15 2012

A(n,k) is the number of partitions of k*(k*n+1)/2 into k distinct parts <=k*n.

A(n,k) = 0 if k>0 and (n = 0 or k*(k*n+1) mod 2 = 1).

			A(0,0) = 1: {}.
A(1,1) = 1: {1}.
A(5,1) = 1: {3}.
A(1,5) = 1: {1,2,3,4,5}.
A(2,2) = 2: {1,4}, {2,3}.
A(3,2) = 3: {1,6}, {2,5}, {3,4}.
A(2,3) = 0: no subset of {1,2,3,4,5,6} has element sum 3*(3*2+1)/2 = 21/2.
A(4,2) = 4: {1,8}, {2,7}, {3,6}, {4,5}.
A(3,3) = 8: {1,5,9}, {1,6,8}, {2,4,9}, {2,5,8}, {2,6,7}, {3,4,8}, {3,5,7}, {4,5,6}.
A(2,4) = 8: {1,2,7,8}, {1,3,6,8}, {1,4,5,8}, {1,4,6,7}, {2,3,5,8}, {2,3,6,7}, {2,4,5,7}, {3,4,5,6}.
Square array A(n,k) begins:
  1, 0, 0,  0,   0,    0,     0,      0, ...
  1, 1, 1,  1,   1,    1,     1,      1, ...
  1, 0, 2,  0,   8,    0,    58,      0, ...
  1, 1, 3,  8,  33,  141,   676,   3370, ...
  1, 0, 4,  0,  86,    0,  3486,      0, ...
  1, 1, 5, 25, 177, 1394, 11963, 108108, ...
  1, 0, 6,  0, 318,    0, 32134,      0, ...
  1, 1, 7, 50, 519, 5910, 73294, 957332, ...

Alois P. Heinz, Antidiagonals d=0..60

Rows n=0-10 give: A000007, A000012, A063074, A109655, A204460, A204461, A204462, A204463, A204464, A204465, A204466.
Columns k=0..10 give: A000012, A000035, A001477, A204467, A204468, A204469, A204470, A204471, A204472, A204473, A204474.
Main diagonal gives: A052456.

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_, i_, t_] /; it*((2*i-t+1)/2) = 0; b[0, , ] = 1; b[n_, i_, t_] := b[n, i, t] = b[n, i-1, t] + If[n, 0] = 1; a[0, ] = 0; a[n_, k_] := With[{s = k*(k*n+1)}, If[Mod[s, 2] == 1, 0, b[s/2, k*n, k]]]; Flatten[ Table[ a[n, d-n], {d, 0, 15}, {n, 0, d}]] (* Jean-François Alcover, Jun 15 2012, translated from Maple, after Alois P. Heinz *)

A225345 T(n,k) = Number of n X k {-1,1}-arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute k-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 0, 3, 6, 7, 0, 0, 1, 0, 9, 0, 15, 0, 1, 0, 3, 12, 31, 0, 33, 8, 0, 1, 0, 17, 0, 107, 0, 77, 0, 1, 0, 5, 22, 81, 0, 395, 410, 181, 0, 0, 1, 0, 27, 0, 397, 0, 1525, 0, 443, 0, 1, 0, 5, 34, 171, 0, 2073, 4508, 6095, 0, 1113, 58, 0, 1, 0, 41, 0, 1081, 0

Offset: 1

R. H. Hardin, May 05 2013

Table starts
.0...1...0.....1....0......1.....0.......1.....0........1......0........1
.0...1...0.....1....0......1.....0.......1.....0........1......0........1
.0...1...0.....3....0......3.....0.......5.....0........5......0........7
.2...3...6.....9...12.....17....22......27....34.......41.....48.......57
.0...7...0....31....0.....81.....0.....171.....0......309......0......509
.0..15...0...107....0....397.....0....1081.....0.....2399......0.....4675
.0..33...0...395....0...2073.....0....7261.....0....19709......0....45385
.8..77.410..1525.4508..11291.25056...50659.95130...168289.283338...457627
.0.181...0..6095....0..63121.....0..364051.....0..1478059......0..4749875
.0.443...0.24893....0.360909.....0.2676331.....0.13280209......0.50435657

			Some solutions for n=4, k=4
.-1.-1.-1..1...-1.-1..1..1...-1..1..1..1...-1.-1.-1.-1...-1.-1.-1..1
.-1..1..1..1...-1..1..1..1...-1.-1.-1..1....1..1..1..1....1..1..1..1
.-1..1..1..1...-1.-1.-1.-1...-1.-1.-1..1....1..1..1..1...-1.-1.-1..1
.-1.-1.-1..1...-1..1..1..1...-1..1..1..1...-1.-1.-1.-1...-1.-1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..3132

Column 1 is A063074(n/4).
Row 3 is A063196(n/2+1).
Row 4 is A008810(n+1).
Row 5 is A202254(n/2).

Empirical for row n:
n=1: a(n) = a(n-2);
n=2: a(n) = a(n-2);
n=3: a(n) = a(n-2) +a(n-4) -a(n-6);
n=4: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -2*a(n-4) +a(n-5);
n=5: a(n) = 3*a(n-2) -2*a(n-4) -2*a(n-6) +3*a(n-8) -a(n-10);
n=6: [order 26, even n];
n=7: [order 42, even n];
n=8: [order 28];
n=9: [order 58, even n];
n=10: [order 90, even n];
n=11: [order 102, even n];
n=12: [order 66].

A060468 Number of fair distributions (equal sum) of the integers {1,..,4n} between A and B = number of solutions to the equation {+-1 +-2 +- 3 ... +-4*n = 0}.

Original entry on oeis.org

1, 2, 14, 124, 1314, 15272, 187692, 2399784, 31592878, 425363952, 5830034720, 81072032060, 1140994231458, 16221323177468, 232615054822964, 3360682669655028, 48870013251334676, 714733339229024336

Offset: 0

Roland Bacher, Mar 15 2001

			a(1)=2: give either the set {1,4} to A and {2,3} to B or give {2,3} to A and {1,4} to B.

Ray Chandler, Table of n, a(n) for n = 0..834 (terms < 10^1000)
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A025591, A060005, A063865, A063867.

a[n_] := Coefficient[Product[q^(-k) + q^k, {k, 1, 4*n}], q, 0]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Sep 26 2013 *)

a(n) = coefficient of q^0 in Product_{k=1..4*n} (q^(-k) + q^k).

a(n) = A025591(4n) = A063865(4n) = A063867(4n) = 2*A060005(n). Seems to be close to sqrt(3/32Pi)*16^n/sqrt(n^3 + n^2*0.6 + n*0.1385...) and sqrt(n*Pi/2)*A063074(n). - Henry Bottomley, Jul 30 2005

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.

Original entry on oeis.org

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

Offset: 0

Wouter Meeussen, Aug 03 2001

nonn

			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)

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

Cf. A047993, A063074, A063746.

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

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

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

A241810 Number of balanced orbitals over n sectors.

Original entry on oeis.org

1, 1, 0, 0, 2, 6, 0, 6, 8, 36, 0, 88, 58, 376, 0, 1096, 526, 4476, 0, 14200, 5448, 57284, 0, 190206, 61108, 764812, 0, 2615268, 723354, 10499504, 0, 36677626, 8908546, 147110276, 0, 522288944, 113093022

Offset: 0

Peter Luschny, Apr 29 2014

For the combinatorial definitions see A232500. An orbital is balanced if its integral is 0. The integral of an orbital w over n sectors is Sum_{k=1..n} Sum_{i=1..k} w(i) where w(i) are the jumps of the orbital represented by -1, 0, 1.

Cf. A232500, A242087.

np[z_]:=Module[{i,j},For[i=Length[z],i>1&&z[[i-1]]>=z[[i]],i--];For[j=Length[z],z[[j]]<=z[[i-1]],j--];Join[Take[z,i-2],{z[[j]]},Reverse[Drop[ReplacePart[z,z[[i-1]],j],i-1]]]];o=Table[1,{16}];
n=0;f=0;Print[1];Print[1];While[n<16,n++;f=1-f;If[OddQ[f*n],Print[0],p=Join[-Take[o,n],{f},Take[o,n-f]];c=0;Do[If[Accumulate[Accumulate[p]][[-1]]==0,c++];p=np[p],{(2*n+1-f)!/(2*n!^2)}];Print[2*c]];n=n-f]
(* Hans Havermann, May 10 2014 *)

def A241810(n):
    if n == 0: return 1
    A = 0
    T = [0] if is_odd(n) else []
    for i in (1..n//2):
        T.append(-1); T.append(1)
    for p in Permutations(T):
        P = 0; S = 0
        for k in (0..n-1):
            P += p[k]; S += P
        if S == 0: A += 1
    return A
[A241810(n) for n in (0..32)]

a(2*n) = A204459(2, n).
a(2*n+1) = A242087(n).
a(4*n) = A063074(n) = A029895(2*n) = A067059(2*n, 2*n).
a(4*n+2) = 0 for all n (proved by H. Havermann).

More terms from Hans Havermann, May 10 2014

a(35), a(36) from Hans Havermann, May 23 2014

A107110 Square array by antidiagonals where T(n,k) is the number of partitions of k into no more than n parts each no more than n. Visible version of A063746.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 3, 2, 1, 1, 0, 0, 0, 3, 3, 2, 1, 1, 0, 0, 0, 3, 5, 3, 2, 1, 1, 0, 0, 0, 3, 5, 5, 3, 2, 1, 1, 0, 0, 0, 2, 7, 7, 5, 3, 2, 1, 1, 0, 0, 0, 1, 7, 9, 7, 5, 3, 2, 1, 1, 0, 0, 0, 1, 8, 11, 11, 7, 5, 3, 2, 1, 1, 0, 0, 0, 0, 7, 14, 13, 11, 7, 5, 3, 2

Offset: 0

Henry Bottomley, May 12 2005

			Rows start 1,0,0,0,...; 1,1,0,0,0,...; 1,1,2,1,1,0,0,0,...; 1,1,2,3,3,3,3,2,1,1,0,0,0,...; 1,1,2,3,5,5,7,7,8,7,7,5,5,3,2,1,1,0,0,0,...; etc.
T(4,6)=7 since 6 can be written seven ways with no more than 4 parts each no more than 4: 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, or 2+2+1+1.

Henry Bottomley, Partition and composition calculator.

Cf. A063746. Fifth row is A102422.

See A063746 for formulas. T(n, k)=A000041(k) if n>=k. T(n, k)=T(n, n^2-k). T(n, [n^2/2])=A029895(n); T(2n, 2n^2)=A063074(n). Row sums are A000984.

A331545 Triangle of constant term of the symmetric q-binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 3, 0, 3, 0, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 0, 4, 0, 8, 0, 4, 0, 1, 1, 1, 4, 8, 12, 12, 8, 4, 1, 1, 1, 0, 5, 0, 18, 0, 18, 0, 5, 0, 1, 1, 1, 5, 13, 24, 32, 32, 24, 13, 5, 1, 1, 1, 0, 6, 0, 33

Offset: 0

Michael Somos, Jan 19 2020

nonn

Symmetric q-binomial coefficients are based on symmetric q-numbers [n] := (q^n-1/q^n)/(q-1/q).

			Triangle begins:
  n\k| 0 1 2 3 4 5 6 7  ...
  ---+----------------
   0 | 1
   1 | 1 1
   2 | 1 0 1
   3 | 1 1 1 1
   4 | 1 0 2 0 1
   5 | 1 1 2 2 1 1
   6 | 1 0 3 0 3 0 1
   7 | 1 1 3 5 5 3 1 1
   ...

CF. A000982, A001973, A063074, A188181.

T[ n_, k_] := Coefficient[ QBinomial[ n, k, x^2] / x^(k (n - k)) // FunctionExpand // Expand, x, 0];

{T(n, k) = if( k<0 || k>n, 0, polcoeff( prod(j = 1, k, (x^(n+1-j) - x^(-n-1+j))/(x^j - x^(-j))), 0))};

T(2*n, 2*k+1) = 0. T(2*n+1, 3) = A000982(n). T(2*n+1, 5) = A001973(n) if n>=2. T(4*n, 2*n) = A063074(n).

cp's OEIS Frontend

A204459 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of k-element subsets that can be chosen from {1,2,...,k*n} having element sum k*(k*n+1)/2.

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A225345 T(n,k) = Number of n X k {-1,1}-arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute k-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

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A060468 Number of fair distributions (equal sum) of the integers {1,..,4n} between A and B = number of solutions to the equation {+-1 +-2 +- 3 ... +-4*n = 0}.

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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.

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Mathematica

PARI

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A241810 Number of balanced orbitals over n sectors.

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A107110 Square array by antidiagonals where T(n,k) is the number of partitions of k into no more than n parts each no more than n. Visible version of A063746.

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A331545 Triangle of constant term of the symmetric q-binomial coefficients.

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A204459 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of k-element subsets that can be chosen from {1,2,...,kn} having element sum k(k*n+1)/2.