A323777 -id:A323777 - cp's OEIS frontend

A166895 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^(n-k)*n/(n-k), n>=1.

1, 3, 7, 39, 366, 5697, 194881, 16288695, 2430565261, 564615230758, 257227244037248, 319346787227133873, 832952161388710135215, 3382434539389226013260403, 22966972221673234523620345857

Offset: 1

Paul D. Hanna, Nov 23 2009

nonn

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 366*x^5/5 + 5697*x^6/6 +...
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 89*x^5 + 1050*x^6 +...+ A166894(n)*x^n +...

G. C. Greubel, Table of n, a(n) for n = 1..50

Cf. A166894, A209428.

Table[Sum[Binomial[n - k, k]^(n - k) *n/(n - k), {k, 0, Floor[n/2]}], {n, 1, 25}] (* G. C. Greubel, May 27 2016 *)

a(n)=sum(k=0,n\2,binomial(n-k,k)^(n-k)*n/(n-k))

Logarithmic derivative of A166894.

Limit_{n->oo} a(n)^(1/n^2) = (1/r - 1)^((1 - r)^2/(3 - 4*r)) = 1.436094496902535711953511352318447104797138641971237143543..., where r = A323777 = 0.220676041323740696312822269998050167187681031027574... is the root of the equation (1 - 2*r)^(3 - 4*r) = (1 - r)^(2*(1 - r))*r^(1 - 2*r). - Vaclav Kotesovec, Nov 20 2024

A209331 a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, n*k-k^2).

Original entry on oeis.org

1, 1, 2, 7, 86, 1905, 66002, 5218373, 1340847046, 688750226335, 527838995308056, 707409447204872377, 2844096719471817175298, 30274246332924074325724393, 517646331335208169889265781259, 13363896516779950029547538703868509

Offset: 0

Paul D. Hanna, Mar 06 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 0..79

Cf. A209330, A167009, A238696, A209428.

Table[Sum[Binomial[(n-k)^2, n*k-k^2], {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)

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

Equals the antidiagonal sums of triangle A209330(n,k) = C(n^2,n*k).

Limit n->infinity a(n)^(1/n^2) = ((1-r)/r)^((1-r)^2/(3-4*r)) = 1.4360944969025357119535113523184471047971386419..., where r = A323777 = 0.220676041323740696312822269998050167187681031... is the root of the equation (1-2*r)^(3-4*r) = (1-r)^(2-2*r) * r^(1-2*r). - Vaclav Kotesovec, Mar 03 2014

Name corrected by Vaclav Kotesovec, Mar 03 2014

A209428 a(n) = Sum_{k=0..[n/2]} binomial(n-k,k)^(n-k).

Original entry on oeis.org

1, 1, 2, 5, 29, 284, 4423, 146913, 12314170, 1881868883, 442540106327, 198351607585964, 242843144659704443, 641109494638274737567, 2641514784666925880476348, 17914201815999230497003603969, 302266027138470510426936352722523

Offset: 0

Paul D. Hanna, Mar 08 2012

nonn

Equals antidiagonal sums of triangle A209427.

G. C. Greubel, Table of n, a(n) for n = 0..82

Cf. A209427, A209331.

Table[Sum[Binomial[n-k,k]^(n-k),{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Mar 06 2014 *)

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

Limit n->infinity a(n)^(1/n^2) = ((1-r)/r)^((1-r)^2/(3-4*r)) = 1.4360944969025357119535113523184471..., where r = A323777 = 0.220676041323740696312822269998... is the root of the equation (1-2*r)^(3-4*r) = (1-r)^(2-2*r) * r^(1-2*r). - Vaclav Kotesovec, Mar 06 2014

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]]

cp's OEIS Frontend

A166895 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^(n-k)*n/(n-k), n>=1.

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A209331 a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, n*k-k^2).

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A209428 a(n) = Sum_{k=0..[n/2]} binomial(n-k,k)^(n-k).

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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

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.