A227403 -id:A227403 - cp's OEIS frontend

A237421 Decimal expansion of the root of the equation (1-r)^(2r) = r^(2r+1).

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

Offset: 0

Vaclav Kotesovec, Mar 04 2014

			0.6032326837741362...

Vaclav Kotesovec, Table of n, a(n) for n = 0..105

Cf. A206152, A227403, A220359.

Digits:= 140:
v:= convert(fsolve((1-r)^(2*r) = r^(2*r+1), r=1/2), string):
seq(parse(v[n+2]), n=0..120);

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

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

Paul D. Hanna, Feb 04 2012

nonn

Ignoring initial term, equals the logarithmic derivative of A206151.

			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; ...

Seiichi Manyama, Table of n, a(n) for n = 0..47

Cf. A206151, A227403, A237421.

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))}
```

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

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

A306206 a(n) = Sum_{k=0..n} (n^2)!/((n^2-nk)!n!^k).

Original entry on oeis.org

1, 2, 13, 3445, 127028721, 1249195963773451, 5343245431687763366112193, 14729376926426500067331714992293420777, 36332859343341728199556523379140726537646663631786369

Offset: 0

Seiichi Manyama, Jan 29 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A206849, A227403, A306207.

{a(n) = sum(k=0, n, (n^2)!/((n^2-n*k)!*n!^k))}

From Vaclav Kotesovec, Jan 29 2019: (Start)
a(n) ~ 2 * (n^2)! / (n!)^n.
a(n) ~ n^(n^2 - n/2 + 1) / (exp(1/12) * 2^((n-3)/2) * Pi^((n-1)/2)). (End)

A306207 a(n) = Sum_{k=0..n} (n^2)!/((n^2-nk)!k!^n).

Original entry on oeis.org

1, 2, 19, 9745, 768211081, 17406784944114721, 179762725526880242306609281, 1230064011299573560897489169488350806401, 7660929590740297929124296619236388608530015362840364161

Offset: 0

Seiichi Manyama, Jan 29 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A206849, A227403, A306206.

{a(n) = sum(k=0, n, (n^2)!/((n^2-n*k)!*k!^n))}

From Vaclav Kotesovec, Jan 29 2019: (Start)
a(n) ~ (n^2)! / (n! * ((n-1)!)^n).
a(n) ~ exp(n - 1/12) * n^(n^2 - n/2 + 1/2) / (2*Pi)^(n/2). (End)

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A237421 Decimal expansion of the root of the equation (1-r)^(2*r) = r^(2*r+1).

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

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

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A306206 a(n) = Sum_{k=0..n} (n^2)!/((n^2-n*k)!*n!^k).

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A306207 a(n) = Sum_{k=0..n} (n^2)!/((n^2-n*k)!*k!^n).

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Formula

A237421 Decimal expansion of the root of the equation (1-r)^(2r) = r^(2r+1).

A306206 a(n) = Sum_{k=0..n} (n^2)!/((n^2-nk)!n!^k).

A306207 a(n) = Sum_{k=0..n} (n^2)!/((n^2-nk)!k!^n).