A334549 -id:A334549 - cp's OEIS frontend

A376935 Array read by antidiagonals: T(n,k) is the number of 2n X 2k binary matrices with all row sums k and column sums n.

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 20, 90, 20, 1, 1, 70, 1860, 1860, 70, 1, 1, 252, 44730, 297200, 44730, 252, 1, 1, 924, 1172556, 60871300, 60871300, 1172556, 924, 1, 1, 3432, 32496156, 14367744720, 116963796250, 14367744720, 32496156, 3432, 1, 1, 12870, 936369720, 3718394156400, 273957842462220, 273957842462220, 3718394156400, 936369720, 12870, 1

Offset: 0

Andrew Howroyd, Oct 11 2024

T(n,k) is the number of 2*n X 2*k {-1,1} matrices with all rows and columns summing to zero.

			Array begins:
========================================================================
n\k | 0   1       2           3               4                   5 ...
----+------------------------------------------------------------------
  0 | 1   1       1           1               1                   1 ...
  1 | 1   2       6          20              70                 252 ...
  2 | 1   6      90        1860           44730             1172556 ...
  3 | 1  20    1860      297200        60871300         14367744720 ...
  4 | 1  70   44730    60871300    116963796250     273957842462220 ...
  5 | 1 252 1172556 14367744720 273957842462220 6736218287430460752 ...
  ...

Nikolai Beluhov, Powers of 2 in Balanced Grid Colourings, arXiv:2504.21451 [math.CO], 2025.
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024.

Main diagonal is A058527.
Columns 0..9 are A000012, A000984, A002896, A172556, A172555, A172557, A172558, A172559, A172560, A172554.
Cf. A008300, A195644, A333901, A334549, A377007 (up to permutations of rows and columns).

T(n, k)={
  local(M=Map(Mat([2*k, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(i, p, v, e) = if(i<0, if(!e, acc(p, v)), my(t=polcoef(p,i)); for(j=0, min(t, e), self()(i-1, p+j*(x-1)*x^i, binomial(t, j)*v, e-j))));
  for(r=1, 2*n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n-1, src[i, 1], src[i, 2], k))); vecsum(Mat(M)[,2]);
}

T(n,k) = T(k,n).

A172634 Number of n X 3 0..2 arrays with row sums 3 and column sums n.

Original entry on oeis.org

1, 1, 7, 31, 175, 991, 5881, 35617, 219871, 1376095, 8710537, 55644337, 358198369, 2320792657, 15120204295, 98984058271, 650725327231, 4293779332927, 28425752310361, 188739799967425, 1256510215733185, 8385127334900305, 56078904057164215, 375796823748323215

Offset: 0

R. H. Hardin, Feb 06 2010

nonn

Inverse binomial transform of the Franel numbers (A000172). - Paul D. Hanna, Feb 26 2012
a(n) is the constant term in the expansion of (1 + x + y + 1/x + 1/y + x/y + y/x)^n. - Seiichi Manyama, Oct 26 2019
a(n) is the constant term in the expansion of (-1 + (1 + x) * (1 + y) + (1 + 1/x) * (1 + 1/y))^n. - Seiichi Manyama, Oct 27 2019
a(n) is the number of n step closed walks on the hexagonal lattice with loops at each node. A step along a loop leaves the position unchanged. The bijection is as follows: after subtracting 1 from each element in the array, values are -1, 0 or 1 and row and column sums are zero. There are only seven possibilities for each row. An all zero row corresponds with a step along the loop leaving the position unchanged and the others to a unit step in each of the six possible directions. This justifies that this sequence is the binomial transform of A002898. - Andrew Howroyd, May 09 2020

			G.f.: A(x) = 1 + x + 7*x^2 + 31*x^3 + 175*x^4 + 991*x^5 + 5881*x^6 +...
G.f.: A(x) = 1/(1-x) + 6*x^2*(1+x)/(1-x)^4 + 90*x^4*(1+x)^2/(1-x)^7 + 1680*x^6*(1+x)^3/(1-x)^10 + 34650*x^8*(1+x)^4/(1-x)^13 +...+ A006480(n)*x^(2*n)*(1+x)^n/(1-x)^(3*n+1) +...

Andrew Howroyd, Table of n, a(n) for n = 0..500 (terms 1..99 from R. H. Hardin)

Column k=3 of A328747 and A334549.

Cf. A000172, A002898, A006480, A328725.

Table[SeriesCoefficient[Sum[(3*k)!/k!^3*x^(2*k)*(1+x)^k/(1-x)^(3*k+1),{k,0,n}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 20 2012 *)

{a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3*x^(2*m)*(1+x)^m/(1-x + x*O(x^n))^(3*m+1)),n)} \\ Paul D. Hanna, Feb 26 2012

a(n)={sum(i=0, n, sum(j=0, i, (-1)^(n-i)*binomial(n, i)*binomial(i, j)^3))} \\ Andrew Howroyd, May 09 2020

From Paul D. Hanna, Feb 26 2012: (Start)
G.f.: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)*(1+x)^n / (1-x)^(3*n+1).
Equals the binomial transform of A002898.
a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(n, k) * A000172(k), where A000172(k) = Sum_{j=0..k} binomial(k,j)^3 forms the Franel numbers.
(End)
Recurrence: n^2*a(n) = (2*n-1)^2*a(n-1) + 19*(n-1)^2*a(n-2) + 14*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 7^(n+1)*sqrt(3)/(12*Pi*n). - Vaclav Kotesovec, Oct 20 2012
G.f.: hypergeom([1/3, 1/3],[1],-27*x*(x+1)^2/((1-7*x)^2*(1+2*x)))/((1+2*x)^(1/3)*(1-7*x)^(2/3)). - Mark van Hoeij, May 07 2013

a(0)=1 prepended by Andrew Howroyd, May 09 2020

A377063 Array read by antidiagonals: T(n,k) is the number of {-1,0,1} n X k matrices with all rows and columns summing to zero up to permutations of rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 10, 6, 3, 1, 1, 1, 1, 26, 30, 12, 3, 1, 1, 1, 1, 71, 166, 117, 18, 4, 1, 1, 1, 1, 197, 981, 1421, 345, 30, 4, 1, 1, 1, 1, 554, 5937, 20326, 9691, 1042, 42, 5, 1, 1, 1, 1, 1570, 36646, 307063, 336596, 63076, 2746, 63, 5, 1, 1

Offset: 0

Andrew Howroyd, Oct 14 2024

Columns are not permutable.

Equivalently, the number of n X k 0..2 arrays with row sums k and column sums n up to permutations of rows.

			Array begins:
===================================================
n\k | 0 1 2  3    4      5        6           7 ...
----+----------------------------------------------
  0 | 1 1 1  1    1      1        1           1 ...
  1 | 1 1 1  1    1      1        1           1 ...
  2 | 1 1 2  4   10     26       71         197 ...
  3 | 1 1 2  6   30    166      981        5937 ...
  4 | 1 1 3 12  117   1421    20326      307063 ...
  5 | 1 1 3 18  345   9691   336596    12650093 ...
  6 | 1 1 4 30 1042  63076  5328136   506525279 ...
  7 | 1 1 4 42 2746 369036 76292516 18490880339 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..434 (first 29 antidiagonals)

Main diagonal is A377064.
Rows n=0..4 are A000012, A000012, A257520, A377065, A377066.
Columns k=0..4 are A000012, A000012, A008619, A377067, A377068.
Cf. A334549.

A172645 Number of n X n 0..2 arrays with row sums n and column sums n.

Original entry on oeis.org

1, 3, 31, 2371, 1084851, 3680774301, 91358224634433, 17470072106054582211, 26088526624958727703324771, 310419652143758898175543447421953, 29785621316391113552729016416250323294253

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..14

Main diagonal of A334549.

A172635 Number of n X 14 0..2 arrays with row sums 14 and column sums n.

Original entry on oeis.org

1, 616227, 15120204295, 2411829493241299, 358060848206529563811, 75490918169569448369461821, 17949532724881770551236183389177, 4867117722741777809293028167592172435

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..14

Column k=14 of A334549.

A172636 Number of n X 7 0..2 arrays with row sums 7 and column sums n.

Original entry on oeis.org

1, 393, 35617, 7343449, 1509893001, 360255871641, 91358224634433, 24567498558526617, 6887820148172502169, 1998635481349606292593, 596268326872362183077193, 182062424087215419645579529

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..47

Column k=7 of A334549.

A172637 Number of n X 10 0..2 arrays with row sums 10 and column sums n.

Original entry on oeis.org

1, 8953, 8710537, 30189264889, 101471705778601, 431222728237019041, 1998635481349606292593, 10132345044562368521279737, 54649595712626140857519780409, 310419652143758898175543447421953

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..25

Column k=10 of A334549.

A172638 Number of n X 8 0..2 arrays with row sums 8 and column sums n.

Original entry on oeis.org

1, 1107, 219871, 115321795, 59794281891, 37015866368181, 24567498558526617, 17470072106054582211, 13039980402350138484115, 10132345044562368521279737, 8132104885895774387439086637, 6705797924085107855686463036869

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..37

Column k=8 of A334549.

A172640 Number of n X 12 0..2 arrays with row sums 12 and column sums n.

Original entry on oeis.org

1, 73789, 358198369, 8365213132981, 185658731463324901, 5520471267708881730181, 182062424087215419645579529, 6705797924085107855686463036869, 266324891937340036349722115366403541

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..18

Column k=12 of A334549.

A172641 Number of n X 9 0..2 arrays with row sums 9 and column sums n.

Original entry on oeis.org

1, 3139, 1376095, 1849858771, 2435292751411, 3938448319935781, 6887820148172502169, 13039980402350138484115, 26088526624958727703324771, 54649595712626140857519780409, 118790606468949916540947848175709

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..30

Column k=9 of A334549.

cp's OEIS Frontend

A376935 Array read by antidiagonals: T(n,k) is the number of 2*n X 2*k binary matrices with all row sums k and column sums n.

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A172634 Number of n X 3 0..2 arrays with row sums 3 and column sums n.

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A377063 Array read by antidiagonals: T(n,k) is the number of {-1,0,1} n X k matrices with all rows and columns summing to zero up to permutations of rows.

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A172645 Number of n X n 0..2 arrays with row sums n and column sums n.

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A172635 Number of n X 14 0..2 arrays with row sums 14 and column sums n.

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A172636 Number of n X 7 0..2 arrays with row sums 7 and column sums n.

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A172637 Number of n X 10 0..2 arrays with row sums 10 and column sums n.

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A172638 Number of n X 8 0..2 arrays with row sums 8 and column sums n.

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A172640 Number of n X 12 0..2 arrays with row sums 12 and column sums n.

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A172641 Number of n X 9 0..2 arrays with row sums 9 and column sums n.

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A376935 Array read by antidiagonals: T(n,k) is the number of 2n X 2k binary matrices with all row sums k and column sums n.