A228833 -id:A228833 - cp's OEIS frontend

A228832 Triangle defined by T(n,k) = binomial(n*k, k^2), for n>=0, k=0..n, as read by rows.

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, 1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1, 1, 8, 1820, 1307504, 601080390, 40225345056, 69668534468, 231917400, 1, 1, 9, 3060, 4686825, 7307872110, 3169870830126, 96926348578605, 37387265592825, 11969016345, 1

Offset: 0

Paul D. Hanna, Sep 04 2013

Central coefficients are A201555(n) = C(2*n^2,n^2) = A000984(n^2), where A000984 is the central binomial coefficients.

			The triangle of coefficients C(n*k, k^2), n>=k, k=0..n, 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;
1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1;
1, 8, 1820, 1307504, 601080390, 40225345056, 69668534468, 231917400, 1;
1, 9, 3060, 4686825, 7307872110, 3169870830126, 96926348578605, 37387265592825, 11969016345, 1; ...

Paul D. Hanna, Rows 0..30, as a flattened table of n, a(n) for n = 0..495

Cf. A228808 (row sums), A228833 (antidiagonal sums), A135860 (diagonal), A201555 (central terms).
Cf. A229052.
Cf. related triangles: A228904 (exp), A209330, A226234, A228836.

{T(n, k)=binomial(n*k, k^2)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

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]]

A228837 a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, (n-2k)k).

Original entry on oeis.org

1, 1, 2, 5, 38, 597, 14472, 554653, 44421258, 8933194659, 3408672951784, 1984802013951149, 1803179670478111304, 3323206887194925488269, 15156709454119350064982141, 132889643918499982093215167857, 1784438297905511051093397284187186

Offset: 0

Paul D. Hanna, Sep 05 2013

nonn

Equals the antidiagonal sums of triangle A228836.

G. C. Greubel, Table of n, a(n) for n = 0..86

Cf. A228836, A207138.

Cf. variants: A209331, A228833, A123165.

Table[Sum[Binomial[(n-k)^2, (n-2*k)*k],{k,0,Floor[n/2]}],{n,0,15}] (* Vaclav Kotesovec, Sep 05 2013 *)

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

Limit n->infinity a(n)^(1/n^2) = ((1-r)^2/(r*(1-2*r)))^((1-3*r)*(1-r)/(3*(1-2*r))) = 1.36198508972775011599..., where r = 0.195220321930105755... is the root of the equation (1-3*r+3*r^2)^(3*(2*r-1)) = (r*(1-2*r))^(4*r-1) * (1-r)^(4*(r-1)). - Vaclav Kotesovec, added Sep 05 2013, simplified Mar 04 2014

cp's OEIS Frontend

A228832 Triangle defined by T(n,k) = binomial(n*k, k^2), for n>=0, k=0..n, as read by rows.

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

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A228837 a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, (n-2k)k).

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cp's OEIS Frontend

A228832 Triangle defined by T(n,k) = binomial(n*k, k^2), for n>=0, k=0..n, as read by rows.

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PARI

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A228837 a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, (n-2*k)*k).

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Mathematica

PARI

Formula

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

A228837 a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, (n-2k)k).