A245259 -id:A245259 - cp's OEIS frontend

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

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

Vaclav Kotesovec, Jan 27 2019

			0.2206760413237406963128222699980501671876810310275740395417335127215630565...

Cf. A166895, A209331, A209428.

Cf. A220359, A237421, A245259, A323768, A323773, A323778.

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 (1-2r)^(4r-1) * (1-r)^(1-2r) = r^(2r).

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

Vaclav Kotesovec, Jan 27 2019

			0.3663201503052830964087236563781171194011826607210994595491823160184...

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

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)^(2-2r) r^(2r) = 1-2r.

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

Vaclav Kotesovec, Jan 27 2019

			0.36549849821985804457973687544629908832275880696346029501595516768211883674...

Cf. A123165, A206851.

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

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

A245242 a(n) = Sum_{k=0..n} binomial(n^2 - k^2, n*k - k^2).

Original entry on oeis.org

1, 2, 5, 40, 987, 73026, 15656191, 9146092572, 15579632823935, 71399036100619112, 916371430754269894286, 33098484899485154272997507, 3182514246669584511131232330210, 875352526298195795986890973534420721, 650999500319874632196352991280266092913655

Offset: 0

Paul D. Hanna, Jul 14 2014

nonn

			We can illustrate the terms as the row sums of triangle A245243;
triangle A245243(n,k) = C(n^2 - k^2, n*k - k^2) begins:
1;
1, 1;
1, 3, 1;
1, 28, 10, 1;
1, 455, 495, 35, 1;
1, 10626, 54264, 8008, 126, 1;
1, 324632, 10518300, 4686825, 125970, 462, 1;
1, 12271512, 3190187286, 5586853480, 354817320, 1961256, 1716, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..67

Cf. A245243, A227403, A245259.

Table[Sum[Binomial[n^2-k^2,n*k-k^2],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 15 2014 *)

{a(n)=sum(k=0,n,binomial(n^2 - k^2, n*k - k^2))}
for(n=0,20,print1(a(n),", "))

{a(n)=sum(k=0,n,binomial(n^2,n*k)*binomial(n*k,k^2)/binomial(n^2,k^2))}
for(n=0,20,print1(a(n),", "))

a(n) = Sum_{k=0..n} C(n^2, n*k) * C(n*k, k^2) / C(n^2, k^2).
a(n) = Sum_{k=0..n} ((n+k)*(n-k))! / ( (n*(n-k))! * (k*(n-k))! ).
a(n) = Sum_{k=0..n} (n^2 - k^2)! / ( (n^2 - n*k)! * (n*k - k^2)! ).
Limit n->infinity a(n)^(1/n^2) = r^(-(1-r)^2/(2*r)) = 1.65459846190854391888257390278..., where r = 0.37667447497728449846981481128313080857... (see A245259) is the root of the equation r^(2*r-1) = (r+1)^(2*r). - Vaclav Kotesovec, Jul 15 2014

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

Vaclav Kotesovec, Jul 15 2014

			0.78218829428019990122029707592674478018190840396629951687...

Cf. A072034, A220359, A237421, A245259.

Maple
```
evalf(solve(r*log(r/(1-r))=1), 100)
```

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]]

Equals 1/(1+LambertW(exp(-1))).

cp's OEIS Frontend

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

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

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

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Mathematica

A245242 a(n) = Sum_{k=0..n} binomial(n^2 - k^2, n*k - k^2).

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

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Examples

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Programs

Maple

Mathematica

Formula

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

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

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