cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A136505 a(n) = binomial(2^n + 1, n).

Original entry on oeis.org

1, 3, 10, 84, 2380, 237336, 82598880, 99949406400, 422825581068000, 6318976181520699840, 337559127276933693852160, 65182103393445184131620004864, 45946437874792132748338425828443136
Offset: 0

Views

Author

Paul D. Hanna, Jan 01 2008

Keywords

Links

Crossrefs

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), this sequence (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. A136507, A136555.

Programs

  • Magma
    [Binomial(2^n +1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
  • Maple
    A136505:= n-> binomial(2^n+1,n); seq(A136505(n), n=0..20); # G. C. Greubel, Mar 14 2021
  • Mathematica
    Table[Binomial[2^n+1,n], {n,0,15}] (* Vaclav Kotesovec, Jul 02 2016 *)
  • PARI
    {a(n)=polcoeff(sum(i=0,n,(1+2^i*x +x*O(x^n))*log(1+2^i*x +x*O(x^n))^i/i!),n)}
  • Sage
    [binomial(2^n +1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

Formula

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

A136506 a(n) = binomial(2^n + 2, n).

Original entry on oeis.org

1, 4, 15, 120, 3060, 278256, 90858768, 105637584000, 436355999662176, 6431591598617108352, 340881559632021623909760, 65533747894341651530074060800, 46081376018330435634530315478453248
Offset: 0

Views

Author

Paul D. Hanna, Jan 01 2008

Keywords

Links

Crossrefs

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), this sequence (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. A136507, A136555.

Programs

  • Magma
    [Binomial(2^n +2, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
  • Maple
    A136506:= n-> binomial(2^n+2,n); seq(A136506(n), n=0..20); # G. C. Greubel, Mar 14 2021
  • Mathematica
    Table[Binomial[2^n+2,n],{n,0,20}] (* Harvey P. Dale, Jun 20 2011 *)
  • PARI
    {a(n)=polcoeff(sum(i=0,n,(1+2^i*x +x*O(x^n))^2*log(1+2^i*x +x*O(x^n))^i/i!),n)}
  • Sage
    [binomial(2^n +2, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

Formula

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

A054780 Number of n-covers of a labeled n-set.

Original entry on oeis.org

1, 1, 3, 32, 1225, 155106, 63602770, 85538516963, 386246934638991, 6001601072676524540, 327951891446717800997416, 64149416776011080449232990868, 45546527789182522411309599498741023, 118653450898277491435912500458608964207578
Offset: 0

Views

Author

Vladeta Jovovic, May 21 2000

Keywords

Comments

Also, number of n X n rational {0,1}-matrices with no zero rows or columns and with all rows distinct, up to permutation of rows.

Examples

			From _Gus Wiseman_, Dec 19 2023: (Start)
Number of ways to choose n nonempty sets with union {1..n}. For example, the a(3) = 32 covers are:
  {1}{2}{3}  {1}{2}{13}  {1}{2}{123}  {1}{12}{123}  {12}{13}{123}
             {1}{2}{23}  {1}{3}{123}  {1}{13}{123}  {12}{23}{123}
             {1}{3}{12}  {1}{12}{13}  {1}{23}{123}  {13}{23}{123}
             {1}{3}{23}  {1}{12}{23}  {2}{12}{123}
             {2}{3}{12}  {1}{13}{23}  {2}{13}{123}
             {2}{3}{13}  {2}{3}{123}  {2}{23}{123}
                         {2}{12}{13}  {3}{12}{123}
                         {2}{12}{23}  {3}{13}{123}
                         {2}{13}{23}  {3}{23}{123}
                         {3}{12}{13}  {12}{13}{23}
                         {3}{12}{23}
                         {3}{13}{23}
(End)

Links

Crossrefs

Main diagonal of A055154.
Cf. A048291, A088310, A181230, A259763.
Covers with any number of edges are counted by A003465, unlabeled A055621.
Connected graphs of this type are counted by A057500, unlabeled A001429.
This is the covering case of A136556.
The case of graphs is A367863, covering case of A116508, unlabeled A006649.
Binomial transform is A367916.
These set-systems have ranks A367917.
The unlabeled version is A368186.
A006129 counts covering graphs, connected A001187, unlabeled A002494.
A046165 counts minimal covers, ranks A309326.
Cf. A006126, A058891, A059201, A133686, A323816, A323817, A323818, A326754, A367862, A367901.

Programs

  • Mathematica
    Join[{1}, Table[Sum[StirlingS1[n+1, k+1]*(2^k - 1)^n, {k, 0, n}]/n!, {n, 1, 15}]] (* Vaclav Kotesovec, Jun 04 2022 *)
Table[Length[Select[Subsets[Rest[Subsets[Range[n]]],{n}],Union@@#==Range[n]&]],{n,0,4}] (* Gus Wiseman, Dec 19 2023 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(n, k)*binomial(2^(n-k)-1, n)) \\ Andrew Howroyd, Jan 20 2024

Formula

a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*binomial(2^(n-k)-1, n).
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n+1, k+1)*(2^k-1)^n.
G.f.: Sum_{n>=0} log(1+(2^n-1)*x)^n/((1+(2^n-1)*x)*n!). - Paul D. Hanna and Vladeta Jovovic, Jan 16 2008
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jun 04 2022
Inverse binomial transform of A367916. - Gus Wiseman, Dec 19 2023

A132683 a(n) = binomial(2^n + n, n).

Original entry on oeis.org

1, 3, 15, 165, 4845, 435897, 131115985, 138432467745, 525783425977953, 7271150092378906305, 368539102493388126164865, 68777035446753808820521420545, 47450879627176629761462147774626305
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2007

Keywords

Examples

			From _Paul D. Hanna_, Feb 25 2009: (Start)
G.f.: A(x) = 1 + 3*x + 15*x^2 + 165*x^3 + 4845*x^4 + 435897*x^5 + ...
A(x) = 1/(1-x) - log(1-2x)/(1-2x) + log(1-4x)^2/((1-4x)*2!) - log(1-8x)^3/((1-8x)*3!) +- ... (End)

Links

Crossrefs

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), this sequence (1,0), A132684 (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

Programs

  • Magma
    [Binomial(2^n +n, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
  • Maple
    A132683:= n-> binomial(2^n +n,n); seq(A132683(n), n=0..20); # G. C. Greubel, Mar 14 2021
  • Mathematica
    Table[Binomial[2^n+n, n], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)
  • PARI
    a(n)=binomial(2^n+n,n)
  • PARI
    {a(n)=polcoeff(sum(m=0,n,(-log(1-2^m*x))^m/((1-2^m*x +x*O(x^n))*m!)),n)} \\ Paul D. Hanna, Feb 25 2009
  • Sage
    [binomial(2^n +n, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

Formula

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

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

Original entry on oeis.org

1, 4, 21, 220, 5985, 501942, 143218999, 145944307080, 542150225230185, 7398714129087308170, 372134605932348010322571, 69146263065062394421802892300, 47589861944854471977019273909187085
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2007

Keywords

Examples

			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)

Links

Crossrefs

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

Programs

  • Magma
    [Binomial(2^n +n+1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
  • Maple
    A132684:= n-> binomial(2^n +n+1, n); seq(A132684(n), n=0..20); # G. C. Greubel, Mar 14 2021
  • Mathematica
    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)
  • PARI
    {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
  • Sage
    [binomial(2^n +n+1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

Formula

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

Views

Author

Paul D. Hanna, Aug 26 2007

Keywords

Links

Crossrefs

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.

Programs

  • Magma
    [Binomial(2^n+2*n,n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
  • Maple
    A132695:= n-> binomial(2^n +2*n,n); seq(A132685(n), n=0..20); # G. C. Greubel, Mar 14 2021
  • Mathematica
    Table[Binomial[2^n+2n,n],{n,0,20}] (* Harvey P. Dale, Jun 01 2016 *)
  • PARI
    a(n)=binomial(2^n+2*n,n)
  • Sage
    [binomial(2^n+2*n,n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

Formula

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

Views

Author

Paul D. Hanna, Aug 26 2007

Keywords

Links

Crossrefs

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.

Programs

  • Magma
    [Binomial(2^n +2*n +1, n): n in [0..20]]; // G. C. Greubel, Mar 13 2021
  • Mathematica
    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)
  • Sage
    [binomial(2^n +2*n +1, n) for n in (0..20)] # G. C. Greubel, Mar 13 2021

Formula

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

Views

Author

Paul D. Hanna, Aug 26 2007

Keywords

Links

Crossrefs

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.

Programs

  • Magma
    [Binomial(2^n +3*n -1, n): n in [0..20]]; // G. C. Greubel, Mar 13 2021
  • Mathematica
    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)
  • Sage
    [binomial(2^n +3*n -1, n) for n in (0..20)] # G. C. Greubel, Mar 13 2021

Formula

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

Views

Author

Paul D. Hanna, Aug 26 2007

Keywords

Links

Crossrefs

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.

Programs

  • Magma
    [Binomial(2^n +3*n, n): n in [0..20]]; // G. C. Greubel, Mar 13 2021
  • Mathematica
    Table[Binomial[2^n+3n,n],{n,0,20}] (* Harvey P. Dale, Oct 30 2018 *)
  • PARI
    a(n)=binomial(2^n+3*n,n)
  • Sage
    [binomial(2^n +3*n, n) for n in (0..20)] # G. C. Greubel, Mar 13 2021

Formula

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

Views

Author

Paul D. Hanna, Aug 26 2007

Keywords

Links

Crossrefs

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.

Programs

  • Magma
    [Binomial(2^n+3*n+1, n) : n in [0..15]]; // Wesley Ivan Hurt, Nov 20 2014
  • Maple
    A132689:=n->binomial(2^n+3*n+1, n): seq(A132689(n), n=0..15); # Wesley Ivan Hurt, Nov 20 2014
  • Mathematica
    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)
  • Sage
    [binomial(2^n +3*n+1, n) for n in (0..15)] # G. C. Greubel, Feb 15 2021

Formula

a(n) = [x^n] 1/(1-x)^(2^n + 2*n + 2).
Previous Showing 11-20 of 26 results. Next

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