A014070 -id:A014070 - cp's OEIS frontend

A088229 Number of n X n (0,1) matrices with distinct rows.

1, 2, 12, 336, 43680, 24165120, 53981544960, 476410007808000, 16517640193528320000, 2252801478912508197273600, 1212983979979000042023881932800, 2587892965783744956308448364029542400, 21943955209199862746410706867184116563968000

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 03 2003

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A014070.

Table[(2^n)!/(2^n - n)!, {n, 1, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)

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

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jul 02 2016

More terms from Ray Chandler, Nov 06 2003

a(0)=1 prepended and terms a(12) and beyond from Andrew Howroyd, Jan 27 2020

A101346 a(n) = binomial(2^n, n-1).

Original entry on oeis.org

1, 4, 28, 560, 35960, 7624512, 5423611200, 13161885792000, 110859231254749120, 3293259778311548232704, 349928324708588104171703296, 134575849279352109587517966790656, 189165427620415586720308268784807487488, 979739920960712963224129514007339757999308800

Offset: 1

Jorge Coveiro, Dec 25 2004

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A014070. - Paul D. Hanna, Jun 21 2009

Maple
```
seq(binomial(2^n,n-1),n=1..20);
```

Table[Binomial[2^n,n-1], {n,1,15}] (* Vaclav Kotesovec, Jul 02 2016 *)

a(n)=binomial(2^n,n-1) \\ Paul D. Hanna, Jun 21 2009

a(n)=polcoeff(x*sum(k=0,n,2^k*log(1+2^k*x+x*O(x^n))^k/k!),n) \\ Paul D. Hanna, Jun 21 2009

G.f.: A(x) = x*Sum_{n>=0} 2^n*log(1+2^n*x)^n/n!. - Paul D. Hanna, Jun 21 2009

a(n) ~ 2^(n*(n-1)) / (n-1)!. - Vaclav Kotesovec, Jul 02 2016

Terms a(13) and beyond from Andrew Howroyd, Feb 12 2020

A136507 a(n) = Sum_{k=0..n} binomial(2^(n-k) + k, n-k).

Original entry on oeis.org

1, 3, 10, 71, 1925, 203904, 75214965, 94608676477, 409763735870986, 6208539881584781823, 334272186911271376874561, 64832512634295914941490910360, 45811927207957062190019240099653265

Offset: 0

Paul D. Hanna, Jan 01 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..59

Cf. A014070 (C(2^n, n)), A136505 (C(2^n+1, n)), A136506 (C(2^n+2, n)).

Cf. A136508, A136509, A136555.

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

A136507:= n-> add(binomial(2^k +n-k, k), k=0..n); seq(A136507(n), n=0..20); # G. C. Greubel, Mar 14 2021

Table[Sum[Binomial[2^(n-k)+k,n-k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Mar 08 2015 *)

{a(n)=sum(k=0,n,binomial(2^(n-k)+k,n-k))}
for(n=0,16, print1(a(n),", "))

/* a(n) = coefficient of x^n in o.g.f. series: */
{a(n)=polcoeff(sum(i=0,n,1/(1-x-2^i*x^2 +x*O(x^n))*log(1+2^i*x +x*O(x^n))^i/i!),n)}
for(n=0,16, print1(a(n),", "))

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

G.f.: A(x) = Sum_{n>=0} (1 - x - 2^n*x^2)^(-1) * log(1 + 2^n*x)^n/n!.
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016
a(n) = Sum_{k=0..n} A136555(n-k+1, k). - G. C. Greubel, Mar 14 2021

A360693 Number T(n,k) of sets of n words of length n over binary alphabet where the first letter occurs k times; triangle T(n,k), n>=0, n-signum(n)<=k<=n*(n-1)+signum(n), read by rows.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 10, 15, 15, 10, 3, 4, 37, 108, 228, 336, 394, 336, 228, 108, 37, 4, 5, 101, 600, 2150, 5645, 11680, 19752, 27820, 32935, 32935, 27820, 19752, 11680, 5645, 2150, 600, 101, 5, 6, 226, 2490, 14745, 61770, 200529, 535674, 1211485, 2368200

Offset: 0

Alois P. Heinz, Feb 16 2023

T(n,k) is defined for all n >= 0 and k >= 0. The triangle contains only the positive elements.

			T(2,3) = 2: {aa,ab}, {aa,ba}.
T(3,3) = 10: {aab,abb,bbb}, {aab,bab,bbb}, {aab,bba,bbb}, {aba,abb,bbb}, {aba,bab,bbb}, {aba,bba,bbb}, {abb,baa,bbb}, {abb,bab,bba}, {baa,bab,bbb}, {baa,bba,bbb}.
T(4,3) = 4: {abbb,babb,bbab,bbbb}, {abbb,babb,bbba,bbbb}, {abbb,bbab,bbba,bbbb}, {babb,bbab,bbba,bbbb}.
Triangle T(n,k) begins:
  1;
  1, 1;
  .  2, 2,  2;
  .  .  3, 10, 15,  15,  10,    3;
  .  .  .   4, 37, 108, 228,  336,  394,   336,   228,   108,    37,     4;
  .  .  .   .   5, 101, 600, 2150, 5645, 11680, 19752, 27820, 32935, 32935, ...;
  ...

Alois P. Heinz, Rows n = 0..33, flattened

Row sums give A014070.
Column sums give A360695.
Main diagonal T(n,n) gives A154323(n-1) for n>=1.
T(n,n-1) gives A000027(n) for n>=1.
T(2n,2n^2) gives A360702.
Cf. A000290, A057427, A220886 (similar triangle for multisets).

g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i), k), k=0..j))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=n-signum(n)..n*(n-1)+signum(n)))(g(n$3)):
seq(T(n), n=0..6);

g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Binomial[n, i], k], {k, 0, j}]]]];
T[n_] := Table[Coefficient[#, x, i], {i, n - Sign[n], n(n - 1) + Sign[n]}]&[g[n, n, n]];
Table[T[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, May 26 2023, after Alois P. Heinz *)

T(n,k) = T(n,n^2-k).

A093048 a(n) = n minus exponent of 2 in n, with a(0) = 0.

Original entry on oeis.org

0, 1, 1, 3, 2, 5, 5, 7, 5, 9, 9, 11, 10, 13, 13, 15, 12, 17, 17, 19, 18, 21, 21, 23, 21, 25, 25, 27, 26, 29, 29, 31, 27, 33, 33, 35, 34, 37, 37, 39, 37, 41, 41, 43, 42, 45, 45, 47, 44, 49, 49, 51, 50, 53, 53, 55, 53, 57, 57, 59, 58, 61, 61, 63, 58, 65, 65, 67, 66, 69

Offset: 0

Ralf Stephan, Mar 16 2004

nonn

			G.f. = x + x^2 + 3*x^3 + 2*x^4 +  5*x^5 + 5*x^6 + 7*x^7 + 5*x^8 + 9*x^9 + ... - _Michael Somos_, Jan 25 2020

R. J. Mathar, Table of n, a(n) for n = 0..1000

a(n) = n - A007814(n) = A093049(n) + 1, n > 0.
a(n) is the exponent of 2 in A002689(n-1), A014070(n), A060690(n), A075101(n).
See also A084623.

A093048 := proc(n)
    n-A007814(n) ;
end proc: # R. J. Mathar, Jul 24 2014

a[ n_] := If[ n == 0, n - IntegerExponent[n, 2]]; (* Michael Somos, Jan 25 2020 *)

a(n) = if(n<1, 0, if(n%2==0, a(n/2) + n/2 - 1, n))

a(n) = n - valuation(n, 2) \\ Jianing Song, Oct 24 2018

def A093048(n): return n-(~n& n-1).bit_length() if n else 0 # Chai Wah Wu, Jul 07 2022

Recurrence: a(2n) = a(n) + n - 1, a(2n+1) = 2n + 1.
G.f.: Sum_{k>=0} (t*(t^3 + t^2 + 1)/(1 - t^2)^2), with t = x^2^k.
a(n) = Sum_{k=1..n} sign(n mod 2^k). - Wesley Ivan Hurt, May 09 2021

A136470 Triangle T, read by rows, where column 0 of T^m = {C(m*2^n, n), n>=0} for all m.

Original entry on oeis.org

1, 2, 1, 6, 8, 1, 56, 128, 32, 1, 1820, 6048, 2176, 128, 1, 201376, 912128, 419328, 34816, 512, 1, 74974368, 449708544, 249300992, 26198016, 548864, 2048, 1, 94525795200, 739136655360, 477013868544, 59943682048, 1604059136, 8650752, 8192, 1, 409663695276000, 4132411271661568, 3028532448264192, 439222754869248, 14159357935616, 98723430400, 136839168, 32768, 1, 6208116950265950720, 80121787455478857728, 65415571433959456768, 10679727629898088448, 399723620798038016, 3391703461396480, 6141702569984, 2172649472, 131072, 1

Offset: 0

Paul D. Hanna, Dec 31 2007

Column 0 of T^(n+1) = row 2n+1 of square array A136462 defined by: A136462(n,k) = C((n+1)*2^(k-1), k); T^n denotes the n-th matrix power of this triangle T = A136470.

			Triangle T begins:
1;
2, 1;
6, 8, 1;
56, 128, 32, 1;
1820, 6048, 2176, 128, 1;
201376, 912128, 419328, 34816, 512, 1;
74974368, 449708544, 249300992, 26198016, 548864, 2048, 1;
94525795200, 739136655360, 477013868544, 59943682048, 1604059136, 8650752, 8192, 1;
409663695276000, 4132411271661568, 3028532448264192, 439222754869248, 14159357935616, 98723430400, 136839168, 32768, 1; ...
Column 0 of T^m is given by: [T^m](n,0) = C(m*2^n, n) for n>=0.

Cf. columns: A014070, A136471, A136472; A136467 (matrix square-root); A136462.

{T(n,k)=local(M=matrix(n+1,n+1,r,c,binomial(r*2^(c-2),c-1)),P); P=matrix(n+1,n+1,r,c,binomial((r+1)*2^(c-2),c-1));((P~*M~^-1)^2)[n+1,k+1]}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Equals the matrix square of triangle A136467.

Diagonals: T(n+1,n) = 2*4^n; T(n+2,n) = 2*8^n*(2^(n+2) + n-1).

A136636 a(n) = n * C(2*3^(n-1), n) for n>=1.

Original entry on oeis.org

2, 30, 2448, 1265004, 4368213360, 106458751541142, 19173684851378353296, 26413015283743616538733008, 285290979402099025600644272168880, 24601033850235942230699563821233785600080

Offset: 1

Vladeta Jovovic and Paul D. Hanna, Jan 15 2008

Equals column 1 of triangle A136635.

Cf. A136635 (triangle), A014070 (main diagonal), A136393 (column 0), A136637 (row sums), A136638 (antidiagonal sums).

A136636:=n->n*binomial(2*3^(n-1), n); seq(A136636(n), n=1..10); # Wesley Ivan Hurt, Apr 29 2014

Table[n*Binomial[2*3^(n - 1), n], {n, 10}] (* Wesley Ivan Hurt, Apr 29 2014 *)

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

a(n) ~ 2^n * 3^(n*(n-1)) / (n-1)!. - Vaclav Kotesovec, Jul 02 2016

A166995 G.f.: C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)!, a power series in x with integer coefficients.

Original entry on oeis.org

1, 0, 8, 32, 2848, 87808, 97425920, 18364346368, 459757145081856, 468713931103109120, 349620381018764380930048, 1788712998645738038832398336, 46562065744123901943395531497144320

Offset: 0

Paul D. Hanna, Nov 22 2009

nonn

			G.f: C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 +...
The g.f. of A166996 is S(x):
S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)!
S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 +...
where C(x) + S(x) = Sum_{n>=0} C(2^n + n - 1, n)*x^n ... (cf. A060690)
and C(x) - S(x) = Sum_{n>=0} C(2^n, n)*(-x)^n ... (cf. A014070).
Related expansions:
C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 +...
C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 +...

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

Cf. A166996, A166997, A166998, A060690, A014070.

Table[(1/2)*(Binomial[2^n + n - 1, n ] + (-1)^n *Binomial[2^n, n]), {n, 0, 10}] (* G. C. Greubel, May 30 2016 *)

{a(n)=polcoeff(sum(k=0,n,log(1-2^(2*k)*x +x*O(x^n))^(2*k)/(2*k)!),n)}

{a(n)=(binomial(2^n + n-1, n) + (-1)^n*binomial(2^n, n))/2} \\ Paul D. Hanna, Nov 24 2009

a(n) = ( C(2^n + n-1, n) + (-1)^n*C(2^n, n) )/2. - Paul D. Hanna, Nov 24 2009

A166996 G.f.: S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)!, a power series in x with integer coefficients.

Original entry on oeis.org

2, 2, 88, 1028, 289184, 22451552, 112890141568, 50093449805856, 6676830881369059840, 15354513520142235310592, 66620888067382334066280699904, 750203718611121304644623635491840

Offset: 1

Paul D. Hanna, Nov 22 2009

nonn

			G.f.: S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 + ...
The g.f. of A166995 is C(x):
C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)!.
C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 + ...
where C(x) + S(x) = Sum_{n>=0} C(2^n + n - 1, n)*x^n ... (cf. A060690)
and C(x) - S(x) = Sum_{n>=0} C(2^n, n)*(-x)^n ... (cf. A014070).
Related expansions:
C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 + ...
C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 + ...

G. C. Greubel, Table of n, a(n) for n = 1..50

Cf. A166995, A166997, A166998, A060690, A014070.

Table[(1/2)*(Binomial[2^n + n - 1, n ] - (-1)^n *Binomial[2^n, n]), {n, 50}] (* G. C. Greubel, May 30 2016 *)

{a(n)=polcoeff(-sum(k=0,n,log(1-2^(2*k+1)*x +x*O(x^n))^(2*k+1)/(2*k+1)!),n)}

{a(n)=(binomial(2^n + n-1, n) - (-1)^n*binomial(2^n, n))/2} \\ Paul D. Hanna, Nov 24 2009

a(n) = (binomial(2^n + n-1, n) - (-1)^n*binomial(2^n, n) )/2. [Paul D. Hanna, Nov 24 2009]

A166998 G.f.: sqrt(C(x)^2 - S(x)^2) where C(x) = Sum_{n>=0} log(1 - 2^(2n)x)^(2n)/(2n)! and S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)x)^(2n+1)/(2n+1)! are the g.f.s of A166995 and A166996, respectively.

Original entry on oeis.org

1, 0, 6, 28, 2684, 85664, 96848424, 18318978896, 459531493100736, 468613553577122688, 349607028167776160389536, 1788682277200384090414421312, 46561932503015793339090359576558496

Offset: 0

Paul D. Hanna, Nov 22 2009

nonn

			G.f: 1 + 6*x^2 + 28*x^3 + 2684*x^4 + 85664*x^5 + 96848424*x^6 +...
which equals sqrt( C(x)^2 - S(x)^2 ) where
C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 +...
S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 +...
Related expansions:
C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 +...
C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 +...

Cf. A166995, A166996, A166997, A060690, A014070.

{a(n)=polcoeff(sqrt(sum(k=0,n,log(1-2^(2*k)*x +x*O(x^n))^(2*k)/(2*k)!)^2-sum(k=0,n,log(1-2^(2*k+1)*x +x*O(x^n))^(2*k+1)/(2*k+1)!)^2),n)}

G.f.: sqrt([C(x)+S(x)]*[C(x)-S(x)]) where C(x) + S(x) = g.f. of A060690 and C(-x) - S(-x) = g.f. of A014070.

Self-convolution yields A166998.

cp's OEIS Frontend

A088229 Number of n X n (0,1) matrices with distinct rows.

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Mathematica

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A101346 a(n) = binomial(2^n, n-1).

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Maple

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A136507 a(n) = Sum_{k=0..n} binomial(2^(n-k) + k, n-k).

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A360693 Number T(n,k) of sets of n words of length n over binary alphabet where the first letter occurs k times; triangle T(n,k), n>=0, n-signum(n)<=k<=n*(n-1)+signum(n), read by rows.

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Maple

Mathematica

Formula

A093048 a(n) = n minus exponent of 2 in n, with a(0) = 0.

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Maple

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Python

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A136470 Triangle T, read by rows, where column 0 of T^m = {C(m*2^n, n), n>=0} for all m.

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Examples

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Programs

PARI

Formula

A136636 a(n) = n * C(2*3^(n-1), n) for n>=1.

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A166998 G.f.: sqrt(C(x)^2 - S(x)^2) where C(x) = Sum_{n>=0} log(1 - 2^(2n)x)^(2n)/(2n)! and S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)x)^(2n+1)/(2n+1)! are the g.f.s of A166995 and A166996, respectively.