A088229 -id:A088229 - cp's OEIS frontend

A014070 a(n) = binomial(2^n, n).

1, 2, 6, 56, 1820, 201376, 74974368, 94525795200, 409663695276000, 6208116950265950720, 334265867498622145619456, 64832175068736596027448301568, 45811862025512780638750907861652480, 119028707533461499951701664512286557511680

Offset: 0

N. J. A. Sloane

a(n) is the number of n X n (0,1) matrices with distinct rows modulo rows permutations. - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 13 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..60

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), this sequence (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), A132689 (3,1).

Cf. A088229, A136393, A136555.

[Binomial(2^n, n): n in [0..25]]; // Vincenzo Librandi, Sep 13 2016

A014070:= n-> binomial(2^n,n); seq(A014070(n), n=0..20); # G. C. Greubel, Mar 14 2021

Table[Binomial[2^n,n],{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011 *)

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

/* G.f. A(x) as Sum of Series: */
a(n)=polcoeff(sum(k=0,n,log(1+2^k*x +x*O(x^n))^k/k!),n) \\ Paul D. Hanna, Dec 28 2007

{a(n) = (1/n!) * sum(k=0,n, stirling(n, k, 1) * 2^(n*k) )}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Feb 05 2023

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

G.f.: A(x) = Sum_{n>=0} log(1 + 2^n*x)^n / n!. - Paul D. Hanna, Dec 28 2007
a(n) = (1/n!) * Sum_{k=0..n} Stirling1(n, k) * 2^(n*k). - Paul D. Hanna, Feb 05 2023
From Vaclav Kotesovec, Jul 02 2016: (Start)
a(n) ~ 2^(n^2) / n!.
a(n) ~ 2^(n^2 - 1/2) * exp(n) / (sqrt(Pi) * n^(n+1/2)).
(End)

A088310 Number of n X n (0,1)-matrices with all rows distinct and all columns distinct.

Original entry on oeis.org

1, 2, 10, 264, 33864, 19158720, 44680224960, 413586858182400, 14960200449325582080, 2109063823453947981680640, 1162864344149083760773678387200, 2520991223487759548686737154649702400, 21598422878151131130336454273775859841843200, 734233037731110118818452425552296701963294284185600

Offset: 0

N. J. A. Sloane, Nov 07 2003

nonn

			a(2) = 10: 00/01, 00/10, 01/00, 01/10, 01/11, 10/00, 10/01, 10/11, 11/01, 11/10.

G. C. Greubel, Table of n, a(n) for n = 0..57

Cf. A088229, A088309.

Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

A088310:= func< n | Factorial(n)*(&+[Binomial(2^k,n)*StirlingFirst(n,k): k in [0..n]]) >;
[A088310(n): n in [0..30]]; // G. C. Greubel, Dec 14 2022

Table[n!*Sum[StirlingS1[n, k]*Binomial[2^k,n], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)

@CachedFunction
def A088310(n): return (-1)^n*factorial(n)*sum((-1)^k*binomial(2^k,n)*stirling_number1(n,k) for k in (0..n))
[A088310(n) for n in range(31)] # G. C. Greubel, Dec 14 2022

a(n) = n! * Sum_{k=0..n} Stirling1(n, k)*binomial(2^k, n). - Vladeta Jovovic, Nov 07 2003
a(n) = Sum_{i=0..n} Sum_{j=0..n} stirling1(n, i) * stirling1(n, j) * 2^(i*j). - Max Alekseyev, Nov 07 2003
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jul 02 2016
a(n) = A181230(n,n).

Suggested by Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 06 2003

a(0)-a(5) from W. Edwin Clark, Nov 07 2003

A088309 Number of equivalence classes of n X n (0,1)-matrices with all rows distinct and all columns distinct.

Original entry on oeis.org

1, 2, 5, 44, 1411, 159656, 62055868, 82060884560, 371036717493194, 5812014504668066528, 320454239459072905856944, 63156145369562679089674952768, 45090502574837184532027563736271152, 117910805393665959622047902193019284914432, 1139353529410754170844431642119963019965901238144

Offset: 0

N. J. A. Sloane, Nov 07 2003

nonn

Two such matrices are equivalent if they differ just by a permutation of the rows.

			a(2) = 5: 00/01, 00/10, 01/10, 01/11, 10/11.

G. C. Greubel, Table of n, a(n) for n = 0..59
G. Kilibarda and V. Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.

Cf. A088229, A088310, A088616.
Main diagonal of A059084.
Binary matrices with distinct rows and columns, various versions: A059202, this sequence, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763.

A088309:= func< n | (&+[Binomial(2^k,n)*StirlingFirst(n,k): k in [0..n]]) >;
[A088309(n): n in [0..30]]; // G. C. Greubel, Dec 15 2022

A088309[n_]:= A088309[n]=Sum[Binomial[2^j,n]*StirlingS1[n,j], {j,0,n}];
Table[A088309[n], {n,0,30}] (* G. C. Greubel, Dec 15 2022 *)

a(n) = sum(k=0, n, stirling(n, k, 1)*binomial(2^k, n)); \\ Michel Marcus, Dec 16 2022

@CachedFunction
def A088309(n): return (-1)^n*sum((-1)^k*binomial(2^k, n)*stirling_number1(n, k) for k in (0..n))
[A088309(n) for n in range(31)] # G. C. Greubel, Dec 15 2022

a(n) = Sum_{k=0..n} Stirling1(n, k)*binomial(2^k, n). - Vladeta Jovovic, Nov 07 2003

a(n) = A088310(n) / n!.

Suggested by Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 06 2003

a(0)-a(5) from W. Edwin Clark, Nov 07 2003

A101925 a(n) = A005187(n) + 1.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 11, 12, 16, 17, 19, 20, 23, 24, 26, 27, 32, 33, 35, 36, 39, 40, 42, 43, 47, 48, 50, 51, 54, 55, 57, 58, 64, 65, 67, 68, 71, 72, 74, 75, 79, 80, 82, 83, 86, 87, 89, 90, 95, 96, 98, 99, 102, 103, 105, 106, 110, 111, 113, 114, 117, 118, 120, 121, 128, 129

Offset: 0

Ralf Stephan, Dec 28 2004

nonn

Exponent of 2 in the sequences A032184, A052278, A060055, A066318, A088229, A101926.

p(n) sequence for k=2, s=0. p(n) = min(j: A046699(j) = n). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences
B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes
B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes, Electronic Journal of Combinatorics, 13 (2006), #R26, 13 pages.
F. Ruskey, C. Deugau, The Combinatorics of Certain k-ary Meta-Fibonacci Sequences, JIS 12 (2009) 09.4.3. [This is a later version than that in the GenMetaFib.html link]

Bisection of A089279. First differences are in A001511.
Cf. A000120, A005187, A046699.
Cf. A032184, A052278, A060055, A066318, A088229, A101926.

Table[IntegerExponent[(2 n)!, 2] + 1, {n, 0, 65}] (* or *)
Table[2 n - DigitCount[2 n, 2, 1] + 1, {n, 0, 65}] (* Michael De Vlieger, Feb 04 2017 *)

PARI
```
a(n)=1+sum(k=1, n, valuation(k,2)+1)
```

a(n)=if(n==0,1,if((n%2)==0,2*a(n/2)+subst(Pol(binary(n)),x,1)-1,a(n-1)+1))

a(n)=2*n+1-hammingweight(n) \\ Charles R Greathouse IV, Dec 29 2022
(Python 3.10+)
def A101925(n): return (n<<1)-n.bit_count()+1 # Chai Wah Wu, Jul 13 2022

Recurrence: a(2n) = 2a(n) + A000120(n) - 1, a(2n+1) = a(2n) + 1.

G.f.: (1 / 1-z) * (z + z * sum(z^(2^i) * (s + (1 / (1 - z^(2^k)))),i=0..infinity)). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

cp's OEIS Frontend

A014070 a(n) = binomial(2^n, n).

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A088310 Number of n X n (0,1)-matrices with all rows distinct and all columns distinct.

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A088309 Number of equivalence classes of n X n (0,1)-matrices with all rows distinct and all columns distinct.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A101925 a(n) = A005187(n) + 1.

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Mathematica

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