A323773 -id:A323773 - cp's OEIS frontend

A323768 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^k.

1, 1, 2, 3, 5, 14, 43, 171, 1234, 9075, 94295, 1685324, 28688843, 804627839, 34189166176, 1379425012899, 106952499421507, 10394354507270548, 1052079100669253203, 221582922117645427461, 48152920476428200426258, 13152336142340905111739041

Offset: 0

Vaclav Kotesovec, Jan 27 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..123

Cf. A323769, A295612.

Table[Sum[Binomial[n-k, k]^k, {k, 0, n/2}], {n, 0, 25}]

{a(n) = sum(k=0, n\2, binomial(n-k, k)^k)} \\ Seiichi Manyama, Jan 27 2019

Limit_{n->infinity} a(n)^(1/n^2) = ((1-r)/r)^(r^2/(4*r-1)) = 1.17123387669321050316385592324128471190583619526359450226558587879190245..., where r = A323773 = 0.3663201503052830964087236563781171194011826607210994595... is the root of the equation (1-2*r)^(4*r-1) * (1-r)^(1-2*r) = r^(2*r).

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

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]]

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]]

A228833 a(n) = Sum_{k=0..[n/2]} binomial((n-k)*k, k^2).

Original entry on oeis.org

1, 1, 2, 3, 5, 20, 77, 437, 5509, 54475, 1031232, 31874836, 789351469, 47552777430, 3302430043985, 223753995897916, 39177880844093733, 5954060239110086680, 1226026438114057710320, 551315671593483499670137, 188615011023291125237647365, 124995445742889226418307452940

Offset: 0

Paul D. Hanna, Sep 04 2013

nonn

Equals antidiagonal sums of triangle A228832.

Cf. A228832.

Table[Sum[Binomial[(n-k)*k, k^2],{k,0,Floor[n/2]}],{n,0,15}] (* Vaclav Kotesovec, Sep 06 2013 *)

{a(n)=sum(k=0,n\2,binomial(n*k-k^2, k^2))}
for(n=0,30,print1(a(n),", "))

Limit n->infinity a(n)^(1/n^2) = ((1-r)/(1-2*r))^(r/2) = 1.171233876693210503..., where r = A323773 = 0.366320150305283... is the root of the equation (1-2*r)^(4*r-1) * (1-r)^(1-2*r) = r^(2*r). - Vaclav Kotesovec, Sep 06 2013

cp's OEIS Frontend

A323768 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^k.

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

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

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A228833 a(n) = Sum_{k=0..[n/2]} binomial((n-k)*k, k^2).

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cp's OEIS Frontend

A323768 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^k.

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Author

Keywords

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Mathematica

PARI

Formula

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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Author

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Examples

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Mathematica

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

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A228833 a(n) = Sum_{k=0..[n/2]} binomial((n-k)*k, k^2).

Views

Author

Keywords

Comments

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Programs

Mathematica

PARI

Formula

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

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