A014062 -id:A014062 - cp's OEIS frontend

A276454 a(n) = A276452(n) + A276451(n) + A276449(n).

1, 2, 22, 464, 13302, 487152, 21475652, 1106550392, 65221981530, 4327577893800, 319187492622012, 25904823495240144, 2294089575287710984, 220132629099295901408, 22751391952803426496488, 2519687900505935894639088, 297684761086123702744203918

Offset: 1

Jason Y.S. Chiu, Hong-Chang Wang, Chiang, Tung-Ying, Sep 03 2016

For a definition and examples of this problem see the comment section of A276449.
The present sequence {a(n)} gives the number of all orbits under C_4 of 2-colored n X n square grids with n squares of one color.
See A054772(n, k) for the table of these total C_4 orbit numbers for 2-colored grids with any number k from {0,1,...,n^2} of squares of one color. - Wolfdieter Lang, Oct 02 2016

			For n = 4 there are A276449(4) = 4 1-orbits, represented by
   + o o +   o + o o   o o + o   o o o o
   o o o o   o o o +   + o o o   o + + o
   o o o o   + o o o   o o o +   o + + o
   + o o +   o o + o   o + o o   o o o o  .
A276451(4) = 12 2-orbits: one of them is
   + o + o   o o o +
   o o o o   + o o o
   o o o o   o o o +
   o + o +   + o o o  ,
and one can take the first one as representative.
A276452(4) = 448 4-orbits: one of them is represented by
   + + + +
   o o o o
   o o o o
   o o o o .
The complete orbit structure for n=4 is 1^4 2^12 4^448, see A276449(4) = 4, A276451(4) = 12, A276452(4) = 448.
a(4) = 448 + 12 + 4 = 464.
A014062(4) = 448*4 + 12*2 + 4*1 = 1820.

Hong-Chang Wang, Table of n, a(n) for n = 1..70

Cf. A014062, A054772, A276451, A276452, A276454.

f[n_] := If[MemberQ[{2, 3}, #], 0, Function[i, Binomial[(2 i) (2 i + #), i]]@ Floor[n/4]] &@ Mod[n, 4];g[n_] := (Function[j, Binomial[2 j (j + Boole@ OddQ@ n), j]]@ Floor[n/2] - f@ n)/2; Table[(Binomial[n^2, n] - 2 g@ n - f@ n)/4 + (Function[j, Binomial[2 j (j + Boole@ OddQ@ n), j]]@ Floor[n/2] - f@ n)/2 + f@ n, {n, 17}] (* Michael De Vlieger, Sep 12 2016 *)

from math import comb as binomial
for j in range(1, 20):
    t = binomial(j * j, j)
    i = j // 2
    if j % 2 == 0:
        d = binomial(2 * i * i, i)
    else:
        d = binomial(2 * i * (i + 1), i)
    a = (t - d) // 4
    if j % 4 == 0:
        c = binomial((j * j // 4), (j // 4))
    elif j % 4 == 1:
        c = binomial(((j - 1) // 2) * ((j - 1) // 2 + 1), ((j - 1) // 4))
    else:
        c = 0
    b = (d - c) // 2
    print(str(j) + " " + str(a + b + c))

a(n) = A276452(n) + A276451(n) + A276449(n) for n = 1, 2, 3, ...,
A014062(n) = A276452(n)*4 + A276451(n)*2 + A276449(n).
a(n) = A054772(n, 2), n >= 1. - Wolfdieter Lang, Oct 02 2016

Edited by Wolfdieter Lang, Oct 02 2016

A290052 Number of X-rays of n X n binary matrices with exactly n ones.

Original entry on oeis.org

1, 1, 4, 23, 139, 860, 5393, 34142, 217717, 1396346, 8997695, 58205686, 377775385, 2458841504, 16043226825, 104901986083, 687221188145, 4509605878736, 29636894936761, 195035340954186, 1285062484293880, 8476508261617168, 55969236979211755, 369900194873712830

Offset: 0

Alois P. Heinz, Jul 19 2017

nonn

The X-ray of a matrix is defined as the sequence of antidiagonal sums.

			a(0) = 1: [].
a(1) = 1: 1.
a(2) = 4: 011, 020, 101, 110.
a(3) = 23: 00021, 00111, 00120, 00201, 00210, 00300, 01011, 01020, 01101, 01110, 01200, 02001, 02010, 02100, 10011, 10020, 10101, 10110, 10200, 11001, 11010, 11100, 12000.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
Index entries for sequences related to binary matrices

Main diagonal of A290057.

Cf. A019589, A014062, A290133.

b:= proc(n, i, t) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
      add(b(n-j, i-t, 1-t), j=0..min(i, n)))))(i*(i+1-t))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..30);

b[n_, i_, t_] := b[n, i, t] = Function[m, If[n > m, 0, If[n == m, 1, Sum[b[n - j, i - t, 1 - t], {j, 0, Min[i, n]}]]]][i*(i + 1 - t)];
a[n_] := b[n, n, 1];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)

A019589(n) <= a(n) <= A014062(n).

a(n) ~ c * 3^(3*n) / (2^(2*n) * sqrt(n)), where c = 0.153294749730773567280925277269616968259180871352428154276351832424636097919... - Vaclav Kotesovec, Jul 22 2017

A081369 Binomial(n^2, n) reduced mod n^2.

Original entry on oeis.org

0, 2, 3, 12, 5, 12, 7, 56, 9, 40, 11, 108, 13, 84, 90, 240, 17, 144, 19, 80, 315, 220, 23, 72, 25, 312, 27, 560, 29, 0, 31, 992, 759, 544, 770, 720, 37, 684, 1053, 520, 41, 252, 43, 1408, 1125, 1012, 47, 1872, 49, 1200, 918, 624, 53, 1404, 2475, 2744, 2223, 1624, 59

Offset: 1

Labos Elemer, Mar 20 2003

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A014062, A081370.

A081369:=n->binomial(n^2, n) mod n^2; seq(A081369(n), n=1..100); # Wesley Ivan Hurt, Jan 28 2014

Table[Mod[Binomial[n^2, n], n^2], {n, 100}] (* Wesley Ivan Hurt, Jan 28 2014 *)

a(n)=Mod[C(n^2, n), n^2]

A258371 Triangle read by rows: T(n,k) is number of ways of arranging n indistinguishable points on an n X n square grid such that k rows contain at least one point.

Original entry on oeis.org

1, 2, 4, 3, 54, 27, 4, 408, 1152, 256, 5, 2500, 22500, 25000, 3125, 6, 13830, 315900, 988200, 583200, 46656, 7, 72030, 3709545, 25882780, 40588905, 14823774, 823543, 8, 360304, 39024384, 535754240, 1766195200, 1657012224, 411041792, 16777216

Offset: 1

Adam J.T. Partridge, May 28 2015

Row sums give A014062, n >= 1.
Leading diagonal is A000312, n >= 1.
The triangle t(n,k) = T(n,k)/binomial(n,k) gives the number of ways to place n stones into the k X n grid of squares such that each of the k rows contains at least one stone. See A259051. One can use a partition array for this (and the T(n,k)) problem. See A258152. - Wolfdieter Lang, Jun 17 2015

			The number of ways of arranging eight pawns on a standard chessboard such that two rows contain at least one pawn is T(8,2)=360304.
Triangle T(n,k) begins:
n\k 1      2        3       4        5       6 ...
1:  1
2:  2      4
3:  3     54      27
4:  4    408    1152      256
5:  5   2500   22500    25000     3125
6:  6  13830  315900   988200   583200   46656
...
n = 7:  7  72030 3709545 25882780  40588905 14823774 823543,
n = 8:  8 360304 39024384 535754240 1766195200 1657012224 411041792 16777216.

Giovanni Resta, Table of n, a(n) for n = 1..1830 (first 60 rows)

Cf. A000312, A014062, A258152, A259051.

T[n_,k_]:= Binomial[n,k] * Sum[Multinomial@@ (Last/@ Tally[e]) * Times@@ Binomial[n,e], {e, IntegerPartitions[n, {k}]}]; Flatten@ Table[ T[n,k],{n,9}, {k,n}] (* Giovanni Resta, May 28 2015 *)

T(n,2) = binomial(n,2)*(binomial(2*n,n)-2). - Giovanni Resta, May 28 2015

A295773 a(n) = Sum_{k=0..n} binomial(k^2, k).

Original entry on oeis.org

1, 2, 8, 92, 1912, 55042, 2002834, 87903418, 4514068786, 265401903136, 17575711359576, 1294325676386112, 104913619501093500, 9281271920245432932, 889811788303594625412, 91895379599481072720852, 10170646981621794947354052, 1200909691326112843842751962

Offset: 0

Vaclav Kotesovec, Nov 27 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..338

Cf. A014062, A226391, A295772, A359665.

[&+[Binomial(k^2, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Jan 10 2019

Table[Sum[Binomial[k^2, k], {k, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(k^2, k)); \\ Michel Marcus, Jan 10 2019

a(n) ~ exp(n - 1/2) * n^(n - 1/2) / sqrt(2*Pi).

A298689 G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n^2, n) * x^n / A(x)^( n^2 ).

Original entry on oeis.org

1, 1, 5, 56, 957, 22312, 666666, 24367474, 1051351629, 52144520972, 2915915251326, 181227240764128, 12382862552065170, 922234506009645794, 74345308066436693828, 6449466281781165675666, 599083515375854753327365, 59328642583049975996828036, 6240245388930730524658068558, 694754212357547941002786433000, 81628078642468462576697539116234

Offset: 0

Paul D. Hanna, Jan 24 2018

nonn

Compare to: Sum_{n>=0} C(m*n,n) * x^n / (1+x)^(m*n) = (1+x)/(1 - (m-1)*x) holds for fixed m.

			G.f.: A(x) = 1 + x + 5*x^2 + 56*x^3 + 957*x^4 + 22312*x^5 + 666666*x^6 + 24367474*x^7 + 1051351629*x^8 + 52144520972*x^9 + 2915915251326*x^10 + 181227240764128*x^11 + 12382862552065170*x^12 + ...
such that
A(x) = 1 + C(1,1)*x/A(x) + C(4,2)*x^2/A(x)^4 + C(9,3)*x^3/A(x)^9 + C(16,4)*x^4/A(x)^16 + C(25,5)*x^5/A(x)^25 + C(36,6)*x^6/A(x)^36 + C(49,7)*x^7/A(x)^49 + ...
more explicitly,
A(x) = 1 + x/A(x) + 6*x^2/A(x)^4 + 84*x^3/A(x)^9 + 1820*x^4/A(x)^16 + 53130*x^5/A(x)^25 + 1947792*x^6/A(x)^36 + 85900584*x^7/A(x)^49 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..260

Cf. A014062, A298690, A298691.

{a(n) = my(A=[1]); for(i=1,n, A = Vec(sum(m=0,#A,binomial(m^2,m) * x^m/Ser(A)^(m^2) ))); A[n+1]}
for(n=0,30,print1(a(n),", "))

a(2^k) is odd for k>=0, and a(n) is even elsewhere except at n=0 (conjecture).

A298695 G.f.: Sum_{n>=0} binomial(n^2, n) * x^n / (1 + x)^(n^2).

Original entry on oeis.org

1, 1, 5, 61, 1123, 27671, 853411, 31603447, 1365807689, 67469763889, 3749935785301, 231591200859701, 15733654527061483, 1166102347943957815, 93629607937879486019, 8096167402408961507311, 750088483178476669111441, 74127049788588758257392161, 7783440821906363883725443813, 865349148215025766722403077229, 101553078711812924877087765912371

Offset: 0

Paul D. Hanna, Feb 04 2018

nonn

Compare g.f. to: Sum_{n>=0} binomial(m*n, n) * x^n / (1+x)^(m*n) = (1+x)/(1 - (m-1)*x) holds for fixed m.

			G.f.: A(x) = 1 + x + 5*x^2 + 61*x^3 + 1123*x^4 + 27671*x^5 + 853411*x^6 + 31603447*x^7 + 1365807689*x^8 + 67469763889*x^9 + 3749935785301*x^10 + ...
such that
A(x) = 1 + C(1,1)*x/(1+x) + C(4,2)*x^2/(1+x)^4 + C(9,3)*x^3/(1+x)^9 + C(16,4)*x^4/(1+x)^16 + C(25,5)*x^5/(1+x)^25 + C(36,6)*x^6/(1+x)^36 + ...
more explicitly,
A(x) = 1 + x/(1+x) + 6*x^2/(1+x)^4 + 84*x^3/(1+x)^9 + 1820*x^4/(1+x)^16 + 53130*x^5/(1+x)^25 + 1947792*x^6/(1+x)^36 + ... + A014062(n)*x^n/(1+x)^(n^2) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A298696, A014062.

terms = 21; s = Sum[Binomial[n^2, n]*x^n/(1 + x)^(n^2), {n, 0, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Feb 06 2018 *)

{a(n) = my(A = sum(m=0,n,binomial(m^2,m)*x^m/(1+x +x*O(x^n))^(m^2) ) ); polcoeff(A,n)}
for(n=0,25, print1(a(n),", "))

a(n) ~ c * d^n * (n-1)!, , where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.176554609483480358231680164050876553672889794... and c = 0.127903391767990118250352331247574466909912463001514793015830303493876... - Vaclav Kotesovec, Oct 10 2020

c = exp(LambertW(-2*exp(-2))^2/8 - 1/2) / (2*Pi*sqrt(1 + LambertW(-2*exp(-2)))). - Vaclav Kotesovec, Mar 18 2022

A107446 a(n) = binomial(n^4, n).

Original entry on oeis.org

1, 1, 120, 85320, 174792640, 782083984500, 6505247592703944, 90471680541391718800, 1951589337580920650595840, 61742372998425082372103866380, 2743355077591282538231819720749000, 165382891003629711761140477728569323368, 13151651360462180865959048190677701749268800

Offset: 0

Zak Seidov, May 26 2005

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..100

Cf. A014062 (binomial(n^2, n)).

Mathematica
```
Table[Binomial[n^4, n], {n, 10}]
```

a(n) = {binomial(n^4, n)} \\ Andrew Howroyd, Feb 03 2020

a(n) ~ e^n*n^(3n - 1/2)/sqrt(2*Pi). - Harlan J. Brothers, Aug 05 2023

a(0)=1 prepended and terms a(11) and beyond from Andrew Howroyd, Feb 03 2020

A109901 a(n) = binomial(n^2, n*(n+1)/2).

Original entry on oeis.org

1, 1, 4, 84, 8008, 3268760, 5567902560, 39049918716424, 1118770292985239888, 130276394656770614583240, 61448471214136179596720592960, 117118180539414377821494470432491764, 900390992257782351906806257139068209113040, 27883369051325994219981405855549095749234579210080

Offset: 0

Amarnath Murthy, Jul 14 2005

8*a(2*n+1)^4 = A182010(n) = number of potential group developed cocyclic Hadamard matrices over (the group) Z_{(2*n+1)^2} X Z^2_2 [Baliga, et al., p. 130]. - L. Edson Jeffery, Apr 10 2012

			a(6) = 36!/(21!*15!) = 5567902560.

Andrew Howroyd, Table of n, a(n) for n = 0..50
A. Baliga and K. J. Horadam, Cocyclic Hadamard matrices over Z_t X Z^2_2, Australas. J. Combin. 11(1995), 123-134.

Cf. variants: A014062 (C(n^2,n*(n-1))), A135757 (C(n*(n+1),n*(n+1)/2)).

Cf. A182010.

seq(binomial(n^2,n*(n+1)/2),n=0..12); # Emeric Deutsch, Jul 16 2005

Table[Binomial[n^2,(n(n+1))/2],{n,20}] (* Harvey P. Dale, Jun 04 2011 *)

PARI
```
a(n)=binomial(n^2,n*(n+1)/2)
```

a(n) = C(n^2, n*(n+1)/2) = (n^2!)/((n(n+1)/2)!*(n(n-1)/2)!).

a(n) = C(n^2, n*(n-1)/2).

More terms from Emeric Deutsch, Jul 16 2005
Offset changed to 0 (with a(0)=1), and name changed slightly by Paul D. Hanna, Jun 24 2011
Terms a(12) and beyond from Andrew Howroyd, Nov 09 2019

A371306 Least positive number k such that binomial(k^2,k) is divisible by n.

Original entry on oeis.org

1, 2, 2, 3, 4, 2, 3, 6, 8, 4, 5, 3, 4, 3, 5, 6, 6, 8, 8, 4, 3, 5, 5, 6, 9, 4, 16, 3, 8, 5, 6, 14, 5, 6, 4, 8, 9, 8, 9, 10, 13, 3, 7, 6, 9, 5, 7, 6, 10, 9, 6, 4, 13, 16, 5, 6, 8, 8, 8, 14, 8, 6, 12, 23, 4, 5, 12, 6, 5, 4, 12, 8, 9, 9, 9, 8, 5, 9, 9, 14, 16, 13, 13, 3, 14, 7, 8, 6, 19, 9, 4, 7, 6, 7, 9, 14, 10, 10, 9, 16

Offset: 1

Seiichi Manyama, Mar 23 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A014062.

a(n) = my(k=1); while(binomial(k^2, k)%n>0, k++); k;

cp's OEIS Frontend

A276454 a(n) = A276452(n) + A276451(n) + A276449(n).

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A290052 Number of X-rays of n X n binary matrices with exactly n ones.

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Maple

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A081369 Binomial(n^2, n) reduced mod n^2.

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Maple

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A258371 Triangle read by rows: T(n,k) is number of ways of arranging n indistinguishable points on an n X n square grid such that k rows contain at least one point.

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A295773 a(n) = Sum_{k=0..n} binomial(k^2, k).

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A298689 G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n^2, n) * x^n / A(x)^( n^2 ).

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A298695 G.f.: Sum_{n>=0} binomial(n^2, n) * x^n / (1 + x)^(n^2).

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