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.

Showing 1-10 of 10 results.

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

Original entry on oeis.org

1, 2, 6, 56, 1820, 201376, 74974368, 94525795200, 409663695276000, 6208116950265950720, 334265867498622145619456, 64832175068736596027448301568, 45811862025512780638750907861652480, 119028707533461499951701664512286557511680
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Magma
    [Binomial(2^n, n): n in [0..25]]; // Vincenzo Librandi, Sep 13 2016
  • Maple
    A014070:= n-> binomial(2^n,n); seq(A014070(n), n=0..20); # G. C. Greubel, Mar 14 2021
  • Mathematica
    Table[Binomial[2^n,n],{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011 *)
  • PARI
    a(n)=binomial(2^n,n)
  • PARI
    /* 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
  • PARI
    {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
  • Sage
    [binomial(2^n, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

Formula

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)

A179431 a(n) = binomial(3^(n-1), n).

Original entry on oeis.org

1, 1, 3, 84, 17550, 25621596, 268715232324, 21091830512086620, 12814543323816738705045, 61742372998425082372103866380, 2399699340005498870742886195375900380, 761689137813999393167583510790986701377432464, 1992997938492157367948224731863936229108552184201415196
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2010

Keywords

Comments

Equals column 0 of triangle T=A179430 where column 0 of T^m equals C(m*3^(n-1), n) at row n for n>=0, m>=0.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 84*x^3 + 17550*x^4 + 25621596*x^5 +...
A(x) = 1 + log(1+3*x)/3 + log(1+3^2*x)^2/(3^2*2!) + log(1+3^3*x)^3/(3^3*3!) + log(1+3^4*x)^4/(3^4*4!) +...

Links

Crossrefs

Cf. A179430, A179432, A136393, A179433, A179434.

Programs

  • Mathematica
    Table[Binomial[3^(n-1),n], {n,0,15}] (* Vaclav Kotesovec, Jul 02 2016 *)
  • PARI
    a(n)=binomial(3^(n-1), n)
  • PARI
    /* G.f. A(x) as Sum of Series: */
{a(n)=polcoeff(sum(k=0, n, (1/3)^k*log(1+3^k*x +x*O(x^n))^k/k!), n)}

Formula

G.f.: A(x) = Sum_{n>=0} (1/3)^n * log(1 + 3^n*x)^n / n!.
a(n) ~ 3^(n*(n-1)) / n!. - Vaclav Kotesovec, Jul 02 2016

Extensions

Terms a(11) and beyond from Andrew Howroyd, Apr 13 2021

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

Views

Author

Vladeta Jovovic and Paul D. Hanna, Jan 15 2008

Keywords

Comments

Equals column 1 of triangle A136635.

Crossrefs

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

Programs

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

Formula

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

A179430 Triangular matrix T where column 0 of T^m equals C(m*3^(n-1), n) at row n for n>=0, m>=0.

Original entry on oeis.org

1, 1, 1, 3, 9, 1, 84, 405, 81, 1, 17550, 121500, 32805, 729, 1, 25621596, 247203171, 82255257, 2539107, 6561, 1, 268715232324, 3543210805275, 1382411964132, 53628242751, 199290375, 59049, 1, 21091830512086620, 373203783345533355
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2010

Keywords

Examples

			Triangle T begins:
1;
1, 1;
3, 9, 1;
84, 405, 81, 1;
17550, 121500, 32805, 729, 1;
25621596, 247203171, 82255257, 2539107, 6561, 1;
268715232324, 3543210805275, 1382411964132, 53628242751, 199290375, 59049, 1;
21091830512086620, 373203783345533355, 165018275857291311, 7607829219099993, 36456526295226, 15884240049, 531441, 1; ...
where column 0 of T equals A179431(n) = C(3^(n-1), n):
[1, 1, 3, 84, 17550, 25621596, 268715232324, ...]. ...
Illustrate row n in column 0 of T^m equals C(m*3^(n-1), n) as follows.
Matrix square T^2 begins:
1;
2, 1;
15, 18, 1;
816, 1539, 162, 1;
316251, 833490, 124659, 1458, 1;
873642672, 3060203490, 585411786, 9861183, 13122, 1; ...
where column 0 of T^2 equals A179432(n) = C(2*3^(n-1), n):
[1, 2, 15, 816, 316251, 873642672, 17743125256857, ...]. ...
Matrix cube T^3 begins:
1;
3, 1;
36, 27, 1;
2925, 3402, 243, 1;
1663740, 2667411, 275562, 2187, 1;
6774333588, 14164214850, 1896890076, 21966228, 19683, 1; ...
where column 0 of T^3 equals A136393(n) = C(3^n, n):
[1, 3, 36, 2925, 1663740, 6774333588, 204208594169580, ...].

Crossrefs

Cf. A179431, A179432, A136393, A179433, A179434, variant: A136467.

Programs

  • PARI
    {T(n, k)=local(M=matrix(n+1, n+1, r, c, binomial(r*3^(c-2), c-1)), P); P=matrix(n+1, n+1, r, c, binomial((r+1)*3^(c-2), c-1)); (P~*M~^-1)[n+1, k+1]}

A179432 a(n) = C(2*3^(n-1), n).

Original entry on oeis.org

1, 2, 15, 816, 316251, 873642672, 17743125256857, 2739097835911193328, 3301626910467952067341626, 31698997711344336177849363574320, 2460103385023594223069956382123378560008
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2010

Keywords

Comments

Equals column 0 in the matrix square of triangle T=A179430 where column 0 of T^m equals C(m*3^(n-1), n) at row n for n>=0, m>=0.

Examples

			G.f.: A(x) = 1 + 2*x + 15*x^2 + 816*x^3 + 316251*x^4 +...
A(x) = 1 + 2*log(1+3*x)/3 + 2^2*log(1+3^2*x)^2/(3^2*2!) + 2^3*log(1+3^3*x)^3/(3^3*3!) + 2^4*log(1+3^4*x)^4/(3^4*4!) +...

Crossrefs

Cf. A179430, A179431, A136393, A179433, A179434.

Programs

  • Mathematica
    Table[Binomial[2*3^(n-1),n], {n,0,15}] (* Vaclav Kotesovec, Jul 02 2016 *)
  • PARI
    {a(n)=binomial(2*3^(n-1), n)}
  • PARI
    /* G.f. A(x) as Sum of Series: */
{a(n)=polcoeff(sum(k=0, n, (2/3)^k*log(1+3^k*x +x*O(x^n))^k/k!), n)}

Formula

G.f.: A(x) = Sum_{n>=0} (2/3)^n * log(1 + 3^n*x)^n / n!.
a(n) ~ 2^n * 3^(n*(n-1)) / n!. - Vaclav Kotesovec, Jul 02 2016

A179433 Column 1 of triangle A179430.

Original entry on oeis.org

1, 9, 405, 121500, 247203171, 3543210805275, 373203783345533355, 299059356226224581923626, 1870707073035678423776605220985, 93075349691648156957700437094276630105
Offset: 0

Views

Author

Paul D. Hanna, Jul 21 2010

Keywords

Comments

T=A179430 is a triangular matrix where column 0 of T^m equals C(m*3^(n-1), n) at row n for n>=0, m>=0.

Examples

			G.f.: A(x) = 1 + 9*x + 405*x^2 + 121500*x^3 + 247203171*x^4 +...

Crossrefs

Cf. A179430, A179431, A136393, A179432, A179434.

Programs

  • PARI
    {a(n)=local(M=matrix(n+2, n+2, r, c, binomial(r*3^(c-2), c-1)), P); P=matrix(n+2, n+2, r, c, binomial((r+1)*3^(c-2), c-1)); (P~*M~^-1)[n+2, 2]}

A179434 Row sums of triangle A179430.

Original entry on oeis.org

1, 2, 13, 571, 172585, 357625693, 5248165593907, 566958191345077996, 465798195439736703244606, 2982999334066325867630228374270, 151658307264909973462110073089257457502
Offset: 0

Views

Author

Paul D. Hanna, Jul 21 2010

Keywords

Comments

T=A179430 is a Triangular matrix where column 0 of T^m equals C(m*3^(n-1), n) at row n for n>=0, m>=0.

Crossrefs

Cf. A179430, A179431, A136393, A179432, A179433.

Programs

  • PARI
    {a(n)=local(M=matrix(n+2, n+2, r, c, binomial(r*3^(c-2), c-1)), P); P=matrix(n+2, n+2, r, c, binomial((r+1)*3^(c-2), c-1)); sum(k=0,n,(P~*M~^-1)[n+1, k+1])}

A136635 Triangle, read by rows, where T(n,k) = C(n,k) * C(2^k*3^(n-k), n) for n>=k>=0.

Original entry on oeis.org

1, 3, 2, 36, 30, 6, 2925, 2448, 660, 56, 1663740, 1265004, 353430, 42504, 1820, 6774333588, 4368213360, 1114691760, 139915440, 8561520, 201376, 204208594169580, 106458751541142, 23004238451040, 2630276490960
Offset: 0

Views

Author

Vladeta Jovovic and Paul D. Hanna, Jan 15 2008

Keywords

Comments

Main diagonal is A014070(n) = C(2^n,n).
Column 0 is A136393(n) = C(3^n,n).
Row sums form A136637.
Antidiagonal sums form A136638.

Examples

			Triangle begins:
1;
3, 2;
36, 30, 6;
2925, 2448, 660, 56;
1663740, 1265004, 353430, 42504, 1820;
6774333588, 4368213360, 1114691760, 139915440, 8561520, 201376;
204208594169580, 106458751541142, 23004238451040, 2630276490960, 167150463480, 5562289824, 74974368; ...

Crossrefs

Cf. A014070 (main diagonal), A136393 (column 0), A136636 (column 1), A136637 (row sums), A136638 (antidiagonal sums).

Programs

  • Mathematica
    Flatten[Table[Binomial[n,k]Binomial[2^k 3^(n-k),n],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Dec 13 2012 *)
  • PARI
    {T(n,k)=binomial(n,k)*binomial(2^k*3^(n-k),n)}
  • PARI
    /* Using g.f.: */ {T(n,k)=polcoeff(polcoeff(sum(i=0,n,log(1+3^i*x+2^i*x*y)^i/i!),n,x),k,y)}

Formula

G.f.: A(x,y) = Sum_{n>=0} log(1 + 3^n*x + 2^n*x*y)^n / n!.

A136637 a(n) = Sum_{k=0..n} C(n, k) * C(2^k*3^(n-k), n).

Original entry on oeis.org

1, 5, 72, 6089, 3326498, 12405917044, 336474648380394, 69883583587428350874, 115099747754889610404191160, 1536533057081060754026861201898620, 168527150638482484315370462123098294514192
Offset: 0

Views

Author

Vladeta Jovovic and Paul D. Hanna, Jan 15 2008

Keywords

Comments

Equals row sums of triangle A136635.

Examples

			More generally,
if Sum_{n>=0} log(1 + (p^n + r*q^n)*x )^n / n! = Sum_{n>=0} b(n)*x^n,
then b(n) = Sum_{k=0..n} C(n,k)*r^(n-k) * C(p^k*q^(n-k), n)
(a result due to _Vladeta Jovovic_, Jan 13 2008).

Crossrefs

Cf. A136635 (triangle), A014070 (main diagonal), A136393 (column 0), A136636 (column 1), A136638 (antidiagonal sums).

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]*Binomial[2^k*3^(n-k),n], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)*binomial(2^k*3^(n-k),n))}
  • PARI
    /* Using g.f.: */ {a(n)=polcoeff(sum(i=0,n,log(1+(2^i+3^i)*x)^i/i!),n,x)}

Formula

G.f.: A(x) = Sum_{n>=0} log(1 + (2^n + 3^n)*x )^n / n!.
a(n) ~ 3^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016

A136638 a(n) = Sum_{k=0..[n/2]} C(n-k, k) * C(3^(n-2*k)*2^k, n-k).

Original entry on oeis.org

1, 3, 38, 2955, 1666194, 6775599252, 204212962736426, 47025953519744215608, 84798028785462127288681736, 1219731316443261012339196962784452, 141916030637329352970764084182705691263552
Offset: 0

Views

Author

Vladeta Jovovic and Paul D. Hanna, Jan 15 2008

Keywords

Comments

Equals antidiagonal sums of triangle A136635.

Examples

			More generally, if Sum_{n>=0} log(1 + b*p^n*x + d*q^n*x^2)^n/n! = Sum_{n>=0} a(n)*x^n then a(n) = Sum_{k=0..[n/2]} C(n-k,k)*b^(n-2k)*d^k*C(p^(n-2k)*q^k,n-k).

Crossrefs

Cf. A136635 (triangle), A014070 (main diagonal), A136393 (column 0), A136636 (column 1), A136637 (row sums).

Programs

  • Mathematica
    Table[Sum[Binomial[n-k,k]*Binomial[2^k*3^(n-2*k),n-k], {k, 0, Floor[n/2]}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)
  • PARI
    {a(n)=sum(k=0,n\2,binomial(n-k,k)*binomial(3^(n-2*k)*2^k,n-k))}
  • PARI
    /* Using g.f.: */ {a(n)=polcoeff(sum(i=0,n,log(1+3^i*x+2^i*x^2)^i/i!),n,x)}

Formula

G.f.: A(x) = Sum_{n>=0} log(1 + 3^n*x + 2^n*x^2)^n / n!.
a(n) ~ 3^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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