A060690 -id:A060690 - cp's OEIS frontend

A132684 a(n) = binomial(2^n + n + 1, n).

1, 4, 21, 220, 5985, 501942, 143218999, 145944307080, 542150225230185, 7398714129087308170, 372134605932348010322571, 69146263065062394421802892300, 47589861944854471977019273909187085

Offset: 0

Paul D. Hanna, Aug 26 2007

nonn

			From _Paul D. Hanna_, Feb 25 2009: (Start)
G.f.: A(x) = 1 + 4*x + 21*x^2 + 220*x^3 + 5985*x^4 + 501942*x^5 +...
A(x) = 1/(1-x)^2 - log(1-2x)/(1-2x)^2 + log(1-4x)^2/((1-4x)^2*2!) - log(1-8x)^3/((1-8x)^2*3!) +- ... (End)

G. C. Greubel, Table of n, a(n) for n = 0..50

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), this sequence (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
Cf. A136555.
Cf. A066384. - Paul D. Hanna, Feb 25 2009

[Binomial(2^n +n+1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021

A132684:= n-> binomial(2^n +n+1, n); seq(A132684(n), n=0..20); # G. C. Greubel, Mar 14 2021

Table[Binomial[2^n+n+1,n],{n,0,20}] (* Harvey P. Dale, Nov 10 2011 *)

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

{a(n)=polcoeff(sum(m=0,n,(-log(1-2^m*x))^m/((1-2^m*x +x*O(x^n))^2*m!)),n)} \\ Paul D. Hanna, Feb 25 2009

[binomial(2^n +n+1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

a(n) = [x^n] 1/(1-x)^(2^n + 2).
G.f.: Sum_{n>=0} (-log(1 - 2^n*x))^n / ((1 - 2^n*x)^2*n!). - Paul D. Hanna, Feb 25 2009
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016

A132685 a(n) = binomial(2^n + 2*n, n).

Original entry on oeis.org

1, 4, 28, 364, 10626, 850668, 218618940, 198773423848, 669741609663270, 8493008777332033900, 405943250253048290447028, 72938914603968404495709630360, 49143490709866058459392200362497820

Offset: 0

Paul D. Hanna, Aug 26 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), this sequence (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).

Cf. A136555.

[Binomial(2^n+2*n,n): n in [0..20]]; // G. C. Greubel, Mar 14 2021

A132695:= n-> binomial(2^n +2*n,n); seq(A132685(n), n=0..20); # G. C. Greubel, Mar 14 2021

Table[Binomial[2^n+2n,n],{n,0,20}] (* Harvey P. Dale, Jun 01 2016 *)

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

[binomial(2^n+2*n,n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

a(n) = [x^n] 1/(1-x)^(2^n + n + 1).

A132686 a(n) = binomial(2^n + 2*n + 1, n).

Original entry on oeis.org

1, 5, 36, 455, 12650, 962598, 237093780, 209004408899, 689960224294614, 8639439963148103450, 409865407260324119340236, 73328394245057556170201283726, 49287010273876375495535472789937580

Offset: 0

Paul D. Hanna, Aug 26 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), this sequence (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).

Cf. A136555.

[Binomial(2^n +2*n +1, n): n in [0..20]]; // G. C. Greubel, Mar 13 2021

Table[Binomial[2^n +2*n +1, n], {n,0,20}] (* G. C. Greubel, Mar 13 2021 *)

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

[binomial(2^n +2*n +1, n) for n in (0..20)] # G. C. Greubel, Mar 13 2021

a(n) = [x^n] 1/(1-x)^(2^n + n + 2).

A132687 a(n) = binomial(2^n + 3*n - 1, n).

Original entry on oeis.org

1, 4, 36, 560, 17550, 1370754, 324540216, 267212177232, 822871715492970, 9728874233306696390, 442491588454024774291770, 76919746769405407508866898400, 50743487119356450255156023756871000

Offset: 0

Paul D. Hanna, Aug 26 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), this sequence (3,-1), A132688 (3,0), A132689 (3,1).

Cf. A136555.

[Binomial(2^n +3*n -1, n): n in [0..20]]; // G. C. Greubel, Mar 13 2021

Table[Binomial[2^n+3n-1,n],{n,0,20}] (* Harvey P. Dale, Sep 07 2017 *)

PARI
```
a(n)=binomial(2^n+3*n-1,n)
```

[binomial(2^n +3*n -1, n) for n in (0..20)] # G. C. Greubel, Mar 13 2021

a(n) = [x^n] 1/(1-x)^(2^n + 2*n).

A132688 a(n) = binomial(2^n + 3*n, n).

Original entry on oeis.org

1, 5, 45, 680, 20475, 1533939, 350161812, 280384608504, 847073824772175, 9894081531608130857, 446730013630787463700695, 77328499046923986969058944720, 50891283683781760304442885961988100

Offset: 0

Paul D. Hanna, Aug 26 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), this sequence (3,0), A132689 (3,1).

Cf. A136555.

[Binomial(2^n +3*n, n): n in [0..20]]; // G. C. Greubel, Mar 13 2021

Table[Binomial[2^n+3n,n],{n,0,20}] (* Harvey P. Dale, Oct 30 2018 *)

PARI
```
a(n)=binomial(2^n+3*n,n)
```

[binomial(2^n +3*n, n) for n in (0..20)] # G. C. Greubel, Mar 13 2021

a(n) = [x^n] 1/(1-x)^(2^n + 2*n + 1).

A132689 a(n) = binomial(2^n + 3*n + 1, n).

Original entry on oeis.org

1, 6, 55, 816, 23751, 1712304, 377447148, 294109729200, 871896500955975, 10061777828754031380, 451004941990890693018405, 77739225019650285306412710240, 51039474754930845750609669420261300

Offset: 0

Paul D. Hanna, Aug 26 2007

G. C. Greubel, Table of n, a(n) for n = 0..50

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), this sequence (3,1).

Cf. A136555.

[Binomial(2^n+3*n+1, n) : n in [0..15]]; // Wesley Ivan Hurt, Nov 20 2014

A132689:=n->binomial(2^n+3*n+1, n): seq(A132689(n), n=0..15); # Wesley Ivan Hurt, Nov 20 2014

Table[Binomial[2^n +3n +1, n], {n, 0, 15}] (* Wesley Ivan Hurt, Nov 20 2014 *)

PARI
```
a(n)=binomial(2^n+3*n+1,n)
```

[binomial(2^n +3*n+1, n) for n in (0..15)] # G. C. Greubel, Feb 15 2021

a(n) = [x^n] 1/(1-x)^(2^n + 2*n + 2).

A086675 Number of n X n (0,1)-matrices modulo cyclic permutations of the rows.

Original entry on oeis.org

1, 2, 10, 176, 16456, 6710912, 11453291200, 80421421917440, 2305843009750581376, 268650182136584290872320, 126765060022823052739661424640, 241677817415439249618874010960064512, 1858395433210885261795036719974526548094976

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 27 2003

Also the number of digraphical necklaces with n vertices. A digraphical necklace is defined to be a directed graph that is minimal among all n rotations of the vertices. Alternatively, it is an equivalence class of directed graphs under rotation of the vertices. These are a kind of partially labeled digraphs. - Gus Wiseman, Mar 04 2019

			From _Gus Wiseman_, Mar 04 2019: (Start)
Inequivalent representatives of the a(2) = 10 digraphical necklace edge-sets:
  {}
  {(1,1)}
  {(1,2)}
  {(1,1),(1,2)}
  {(1,1),(2,1)}
  {(1,1),(2,2)}
  {(1,2),(2,1)}
  {(1,1),(1,2),(2,1)}
  {(1,1),(1,2),(2,2)}
  {(1,1),(1,2),(2,1),(2,2)}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..57
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.

Cf. A000031 (binary necklaces), A000939 (cycle necklaces), A008965, A060690, A061417 (permutation necklaces), A184271, A192332 (graphical necklaces), A275527 (path necklaces), A323858 (toroidal necklaces), A323870.

Cf. A324463, A324464, A324513, A324514.

Table[Fold[ #1+EulerPhi[ #2] 2^(n^2 /#2)&, 0, Divisors[n]]/n, {n, 16}]
(* second program *)
rotdigra[g_,m_]:=Sort[g/.k_Integer:>If[k==m,1,k+1]];
Table[Length[Select[Subsets[Tuples[Range[n],2]],#=={}||#==First[Sort[Table[Nest[rotdigra[#,n]&,#,j],{j,n}]]]&]],{n,0,4}] (* Gus Wiseman, Mar 04 2019 *)

a(n) = (1/n)*Sum_{ d divides n } phi(d)*2^(n^2/d) for n > 0, a(0) = 1.

More terms from Wouter Meeussen, Jul 29 2003

a(0)=1 prepended by Gus Wiseman, Mar 04 2019

A137153 Triangle, read by rows, where T(n,k) = C(2^k + n-k-1, n-k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 10, 8, 1, 1, 5, 20, 36, 16, 1, 1, 6, 35, 120, 136, 32, 1, 1, 7, 56, 330, 816, 528, 64, 1, 1, 8, 84, 792, 3876, 5984, 2080, 128, 1, 1, 9, 120, 1716, 15504, 52360, 45760, 8256, 256, 1, 1, 10, 165, 3432, 54264, 376992, 766480, 357760, 32896

Offset: 0

Paul D. Hanna, Jan 24 2008

Matrix inverse is A137156.

T(n,k) is the number of relations between a set of k distinguishable elements and a set of n-k indistinguishable elements. - Isaac R. Browne, Jun 04 2025

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   4,    1;
  1,  4,  10,    8,     1;
  1,  5,  20,   36,    16,      1;
  1,  6,  35,  120,   136,     32,      1;
  1,  7,  56,  330,   816,    528,     64,      1;
  1,  8,  84,  792,  3876,   5984,   2080,    128,     1;
  1,  9, 120, 1716, 15504,  52360,  45760,   8256,   256,   1;
  1, 10, 165, 3432, 54264, 376992, 766480, 357760, 32896, 512, 1;
  ...

Paul D. Hanna, Table of n, a(n) for n = 0..1080; rows 0..45 of flattened triangle.

Cf. A137154 (row sums), A137155 (antidiagonal sums), A060690 (central terms); A137156 (matrix inverse).

Cf. A092056 (same with reflected rows).

Table[Binomial[2^k+n-k-1,n-k],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 06 2017 *)

{T(n,k)=binomial(2^k+n-k-1,n-k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

{T(n, k) = polcoeff(1/(1-x+x*O(x^n))^(2^k), n-k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

A089006 Number of distinct n X n (0,1) matrices after double sorting: by row, by column, by row .. until reaching a fixed point.

Original entry on oeis.org

1, 2, 7, 45, 650, 24520, 2625117, 836488618, 818230288201, 2513135860300849, 24686082394548211147, 787959836124458000837941, 82905574521614049485027140026

Offset: 0

Wouter Meeussen, Nov 03 2003

Also, number of n X n binary matrices with both rows and columns, considered as binary numbers, in nondecreasing order. (Ordering only rows gives A060690.) - R. H. Hardin, May 08 2008
A result of Adolf Mader and Otto Mutzbauer shows that the two definitions are equivalent. - Victor S. Miller, Feb 03 2009
For n=5, only 0.07% remain distinct. Sorting columns and\or rows does not change the permanent of the matrix and leaves the absolute value of the determinant unchanged.
Diagonal of A180985.

			The 7 (2 X 2)-matrices are {{0,0},{0,0}}, {{0,0},{0,1}}, {{0,0},{1,1}}, {{0,1},{0,1}}, {{0,1},{1,0}}, {{0,1},{1,1}} and {{1,1},{1,1}}.

Adolf Mader and Otto Mutzbauer, "Double Orderings of (0,1) Matrices", Ars Combinatoria v. 61 (2001) pp 81-95.

R. H. Hardin, Binary arrays with both rows and cols sorted, symmetries
M. Werner, An Algorithmic Approach for the Zarankiewicz Problem, Slides, 2012. - From _N. J. A. Sloane_, Jan 01 2013

Cf. A088672, A087981, A180985.

Column 0 of A374525.

baseform[li_List] := FixedPoint[Sort[Transpose[Sort[Transpose[Sort[ #1]]]]]&, li]; Table[Length@Split[Sort[baseform/@(Partition[ #, n]&/@(IntegerDigits[Range[0, -1+2^n^2], 2, n^2]))]], {n, 4}]

a(6)-a(12) found by R. H. Hardin, May 08 2008. These terms were found using bdd's (binary decision diagrams), just setting up the logical relations between bits in a gigantic bdd expression and using that to count the satisfying states.

Edited by N. J. A. Sloane, Feb 05 2009 at the suggestion of Victor S. Miller

A092056 Square table read by downward antidiagonals where T(n,k) = binomial(n+2^k-1,n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 10, 4, 1, 1, 16, 36, 20, 5, 1, 1, 32, 136, 120, 35, 6, 1, 1, 64, 528, 816, 330, 56, 7, 1, 1, 128, 2080, 5984, 3876, 792, 84, 8, 1, 1, 256, 8256, 45760, 52360, 15504, 1716, 120, 9, 1, 1, 512, 32896, 357760, 766480, 376992, 54264, 3432, 165, 10, 1

Offset: 0

Henry Bottomley, Feb 19 2004

Each column is convolution of preceding column starting from the all 1's sequence.

T(n,k) is the number of relations between a set of k distinguishable elements and a set of n indistinguishable elements. - Isaac R. Browne, May 14 2025

			Rows start:
  1, 1,  1,   1,    1,     1,      1,...
  1, 2,  4,   8,   16,    32,     64,...
  1, 3, 10,  36,  136,   528,   2080,...
  1, 4, 20, 120,  816,  5984,  45760,...
  1, 5, 35, 330, 3876, 52360, 766480,...
  ...

Columns include (essentially) A000012, A000027, A000292, A000580, A010968, etc.
Rows include A000012, A000079, A007582, A092056.
Main diagonal gives A060690.
Cf. A137153 (same with reflected antidiagonals).

T(n,k) = Sum_{i=0..n} T(i,k-1)*T(n-i,k-1) starting with T(n,0) = 1 for n>=0.

cp's OEIS Frontend

A132684 a(n) = binomial(2^n + n + 1, n).

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A132685 a(n) = binomial(2^n + 2*n, n).

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A132686 a(n) = binomial(2^n + 2*n + 1, n).

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Magma

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A132687 a(n) = binomial(2^n + 3*n - 1, n).

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A132688 a(n) = binomial(2^n + 3*n, n).

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Magma

Mathematica

PARI

Sage

Formula

A132689 a(n) = binomial(2^n + 3*n + 1, n).

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Magma

Maple

Mathematica

PARI

Sage

Formula

A086675 Number of n X n (0,1)-matrices modulo cyclic permutations of the rows.

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