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 1-8 of 8 results.

A220359 Decimal expansion of the root of the equation (1-r)^(2*r-1) = r^(2*r).

Original entry on oeis.org

7, 0, 3, 5, 0, 6, 0, 7, 6, 4, 3, 0, 6, 6, 2, 4, 3, 0, 9, 6, 9, 2, 9, 6, 6, 1, 6, 2, 1, 7, 7, 7, 0, 9, 5, 2, 1, 3, 2, 4, 6, 8, 4, 5, 7, 4, 2, 4, 2, 8, 1, 5, 5, 5, 5, 8, 6, 2, 1, 5, 7, 1, 6, 5, 1, 0, 5, 1, 2, 3, 0, 6, 0, 0, 3, 9, 9, 4, 0, 1, 4, 4, 9, 5, 2, 5, 4, 5, 6, 8, 0, 4, 6, 0, 5, 7, 3, 1, 5, 1, 9, 8, 5, 4, 4, 8, 3
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 12 2012

Keywords

Comments

Constant is associated with A167008, A219206 and A219207.

Examples

			0.70350607643066243...

Crossrefs

Cf. A167008, A219206, A219207, A228808, A184731, A206154, A206156, A206158, A237421, A295611.

Programs

  • Maple
    Digits:= 140:
v:= convert(fsolve( (1-r)^(2*r-1) = r^(2*r), r=1/2), string):
seq(parse(v[n+2]), n=0..120);  # Alois P. Heinz, Dec 12 2012
  • Mathematica
    RealDigits[r/.FindRoot[(1-r)^(2*r-1)==r^(2*r),{r,1/2}, WorkingPrecision->250], 10, 200][[1]]
  • PARI
    solve(x=.7,1,(1-x)^(2*x-1) - x^(2*x)) \\ Charles R Greathouse IV, Apr 25 2016

A206152 a(n) = Sum_{k=0..n} binomial(n,k)^(n+k).

Original entry on oeis.org

1, 2, 10, 326, 64066, 111968752, 1091576358244, 106664423412770932, 67305628532703785062402, 329378455047908259704557301276, 15577435010841058543979449475481629020, 4149966977623235242137197627437116176363522092
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Ignoring initial term, equals the logarithmic derivative of A206151.

Examples

			L.g.f.: L(x) = 2*x + 10*x^2/2 + 326*x^3/3 + 64066*x^4/4 + 111968752*x^5/5 +...
where exponentiation yields A206151:
exp(L(x)) = 1 + 2*x + 7*x^2 + 120*x^3 + 16257*x^4 + 22426576*x^5 +...
Illustration of initial terms:
a(1) = 1^1 + 1^2 = 2;
a(2) = 1^2 + 2^3 + 1^4 = 10;
a(3) = 1^3 + 3^4 + 3^5 + 1^6 = 326;
a(4) = 1^4 + 4^5 + 6^6 + 4^7 + 1^8 = 64066; ...

Links

Crossrefs

Cf. A206151, A227403, A237421.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]^(n+k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)^(n+k))}

Formula

Limit n->infinity a(n)^(1/n^2) = r^(-(1+r)^2/(2*r)) = 2.93544172048274005711865243..., where r = 0.6032326837741362... (see A237421) is the root of the equation (1-r)^(2*r) = r^(2*r+1). - Vaclav Kotesovec, Mar 03 2014

A227403 a(n) = Sum_{k=0..n} binomial(n^2, n*k) * binomial(n*k, k^2).

Original entry on oeis.org

1, 2, 14, 1514, 1308582, 17304263902, 1362702892177706, 1323407909279927430346, 11218363871234340925730020646, 637467717878006909442727527733810142, 519660435252919757259949810325837093364580014, 2289503386759572781844843312201361014103189493095636611
Offset: 0

Views

Author

Paul D. Hanna, Sep 20 2013

Keywords

Examples

			The following triangles illustrate the terms involved in the sum
a(n) = Sum_{k=0..n} A209330(n,k) * A228832(n,k).
Triangle A209330(n,k) = binomial(n^2, n*k) begins:
1;
1, 1;
1, 6, 1;
1, 84, 84, 1;
1, 1820, 12870, 1820, 1;
1, 53130, 3268760, 3268760, 53130, 1;
1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1;
...
Triangle A228832(n,k) = binomial(n*k, k^2) begins:
1;
1, 1;
1, 2, 1;
1, 3, 15, 1;
1, 4, 70, 220, 1;
1, 5, 210, 5005, 4845, 1;
1, 6, 495, 48620, 735471, 142506, 1; ...

Links

Crossrefs

Cf. A209330, A228832, A206849, A206152, A237421.

Programs

  • Mathematica
    Table[Sum[Binomial[n^2,n*k]*Binomial[n*k,k^2],{k,0,n}],{n,0,10}] (* Vaclav Kotesovec, Sep 21 2013 *)
r^(-(1+r)^2/(2*r))/.FindRoot[(1-r)^(2*r) == r^(2*r+1), {r,1/2}, WorkingPrecision->50] (* program for numerical value of the limit n->infinity a(n)^(1/n^2), Vaclav Kotesovec, Sep 21 2013 *)
  • PARI
    {a(n)=sum(k=0, n, binomial(n^2, n*k)*binomial(n*k, k^2))}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} (n^2)! / ( (n^2-n*k)! * (n*k-k^2)! * (k^2)! ).
Limit n->infinity a(n)^(1/n^2) = r^(-(1+r)^2/(2*r)) = 2.93544172048274..., where r = 0.6032326837741362... (see A237421) is the root of the equation (1-r)^(2*r) = r^(2*r+1). - Vaclav Kotesovec, Sep 21 2013

A245259 Decimal expansion of the root of the equation r^(2*r-1) = (r+1)^(2*r).

Original entry on oeis.org

3, 7, 6, 6, 7, 4, 4, 7, 4, 9, 7, 7, 2, 8, 4, 4, 9, 8, 4, 6, 9, 8, 1, 4, 8, 1, 1, 2, 8, 3, 1, 3, 0, 8, 0, 8, 5, 7, 0, 4, 1, 2, 9, 5, 2, 9, 9, 9, 4, 4, 3, 8, 6, 5, 0, 9, 9, 9, 2, 6, 9, 5, 7, 9, 5, 3, 3, 4, 0, 2, 9, 5, 6, 0, 9, 9, 5, 6, 8, 7, 1, 2, 1, 4, 6, 5, 4, 6, 1, 5, 7, 6, 9, 7, 0, 8, 4, 1, 5, 3, 0, 2, 2, 2, 7, 6
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 15 2014

Keywords

Comments

Constant is associated with A245242.

Examples

			0.3766744749772844984698148112831308085704129529994...

Crossrefs

Cf. A245242, A220359, A237421, A245260.

Programs

  • Mathematica
    RealDigits[r/.FindRoot[r^(2*r-1) == (r+1)^(2*r), {r, 1/2}, WorkingPrecision->250], 10, 200][[1]]

A323777 Decimal expansion of the root of the equation (1-2*r)^(3-4*r) = (1-r)^(2-2*r) * r^(1-2*r).

Original entry on oeis.org

2, 2, 0, 6, 7, 6, 0, 4, 1, 3, 2, 3, 7, 4, 0, 6, 9, 6, 3, 1, 2, 8, 2, 2, 2, 6, 9, 9, 9, 8, 0, 5, 0, 1, 6, 7, 1, 8, 7, 6, 8, 1, 0, 3, 1, 0, 2, 7, 5, 7, 4, 0, 3, 9, 5, 4, 1, 7, 3, 3, 5, 1, 2, 7, 2, 1, 5, 6, 3, 0, 5, 6, 5, 0, 5, 8, 5, 2, 2, 8, 6, 0, 3, 0, 9, 2, 1, 4, 9, 8, 9, 2, 1, 2, 8, 3, 0, 9, 2, 4, 6, 0, 5, 3, 4, 7
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 27 2019

Keywords

Examples

			0.2206760413237406963128222699980501671876810310275740395417335127215630565...

Crossrefs

Cf. A166895, A209331, A209428.
Cf. A220359, A237421, A245259, A323768, A323773, A323778.

Programs

  • Mathematica
    RealDigits[r/.FindRoot[(1-2*r)^(3-4*r) == (1-r)^(2-2*r) * r^(1-2*r), {r, 1/3}, WorkingPrecision->250], 10, 200][[1]]

A323773 Decimal expansion of the root of the equation (1-2*r)^(4*r-1) * (1-r)^(1-2*r) = r^(2*r).

Original entry on oeis.org

3, 6, 6, 3, 2, 0, 1, 5, 0, 3, 0, 5, 2, 8, 3, 0, 9, 6, 4, 0, 8, 7, 2, 3, 6, 5, 6, 3, 7, 8, 1, 1, 7, 1, 1, 9, 4, 0, 1, 1, 8, 2, 6, 6, 0, 7, 2, 1, 0, 9, 9, 4, 5, 9, 5, 4, 9, 1, 8, 2, 3, 1, 6, 0, 1, 8, 4, 0, 5, 2, 1, 3, 5, 4, 9, 0, 0, 9, 8, 9, 2, 5, 8, 2, 5, 7, 6, 7, 1, 9, 5, 2, 1, 9, 5, 9, 0, 0, 0, 1, 6, 8, 6, 7, 4, 6
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 27 2019

Keywords

Examples

			0.3663201503052830964087236563781171194011826607210994595491823160184...

Crossrefs

Cf. A220359, A228833, A237421, A245259, A323768, A323777, A323778.

Programs

  • Mathematica
    RealDigits[r/.FindRoot[(1-2*r)^(4*r-1) * (1-r)^(1-2*r) == r^(2*r), {r, 1/3}, WorkingPrecision->250], 10, 200][[1]]

A323778 Decimal expansion of the root of the equation (1-r)^(2-2*r) * r^(2*r) = 1-2*r.

Original entry on oeis.org

3, 6, 5, 4, 9, 8, 4, 9, 8, 2, 1, 9, 8, 5, 8, 0, 4, 4, 5, 7, 9, 7, 3, 6, 8, 7, 5, 4, 4, 6, 2, 9, 9, 0, 8, 8, 3, 2, 2, 7, 5, 8, 8, 0, 6, 9, 6, 3, 4, 6, 0, 2, 9, 5, 0, 1, 5, 9, 5, 5, 1, 6, 7, 6, 8, 2, 1, 1, 8, 8, 3, 6, 7, 4, 0, 8, 4, 8, 7, 3, 0, 0, 3, 5, 2, 2, 8, 4, 1, 0, 7, 4, 0, 8, 2, 1, 5, 4, 8, 5, 3, 8, 7, 5, 7, 8
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 27 2019

Keywords

Examples

			0.36549849821985804457973687544629908832275880696346029501595516768211883674...

Crossrefs

Cf. A123165, A206851.
Cf. A220359, A237421, A245259, A323768, A323773, A323777.

Programs

  • Mathematica
    RealDigits[r/.FindRoot[(1-r)^(2-2*r) * r^(2*r) == 1-2*r, {r, 1/3}, WorkingPrecision->250], 10, 200][[1]]

A245260 Decimal expansion of the root of the equation r*log(r/(1-r))=1.

Original entry on oeis.org

7, 8, 2, 1, 8, 8, 2, 9, 4, 2, 8, 0, 1, 9, 9, 9, 0, 1, 2, 2, 0, 2, 9, 7, 0, 7, 5, 9, 2, 6, 7, 4, 4, 7, 8, 0, 1, 8, 1, 9, 0, 8, 4, 0, 3, 9, 6, 6, 2, 9, 9, 5, 1, 6, 8, 7, 0, 9, 6, 8, 3, 3, 2, 3, 9, 5, 6, 9, 1, 6, 9, 9, 4, 1, 2, 4, 6, 7, 4, 6, 7, 1, 9, 5, 3, 8, 2, 3, 9, 2, 9, 0, 6, 6, 7, 3, 2, 5, 1, 3, 6, 6, 7, 5, 8, 5
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 15 2014

Keywords

Examples

			0.78218829428019990122029707592674478018190840396629951687...

Crossrefs

Cf. A072034, A220359, A237421, A245259.

Programs

  • Maple
    evalf(solve(r*log(r/(1-r))=1), 100)
  • Mathematica
    RealDigits[r/.FindRoot[r*Log[r/(1-r)]==1, {r, 3/4}, WorkingPrecision->250], 10, 200][[1]]
RealDigits[1/(1+LambertW[E^(-1)]), 10, 200][[1]]

Formula

Equals 1/(1+LambertW(exp(-1))).
Showing 1-8 of 8 results.

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