A345876 -id:A345876 - cp's OEIS frontend

A383853 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(4*n).

1, 1, 260, 556032, 4641176128, 106519579045760, 5472276566891956224, 549375993583284180705280, 97867116732573493470161420288, 28783909470167571938915053763592192, 13216052972619446942074113385580542689280, 9058922175695195359062480694771506779050213376

Offset: 0

Vaclav Kotesovec, May 12 2025

nonn

Cf. A209289, A298851, A345876.

Join[{1}, Table[Sum[Binomial[2*n, k]*(n-k)^(4*n), {k, 0, n}], {n, 1, 12}]] (* or *)
Join[{1}, Table[Sum[Binomial[2*n, n+k]*k^(4*n), {k, 0, n}], {n, 1, 12}]]

a(n) = Sum_{k=0..n} binomial(2*n, n+k) * k^(4*n).

a(n) ~ 4^n * r^(4*n+1) * n^(4*n) / (sqrt(2 - r^2) * (1 - r^2)^n * exp(4*n)), where r = 0.9683644349844134852843167967986294187258222293516... is the root of the equation (1+r)/(1-r) = exp(4/r).

A383916 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(3*n).

Original entry on oeis.org

1, 1, 68, 22770, 21143488, 41904629550, 151957171590144, 910666718387157732, 8390164064875701321728, 112583179357513548960803670, 2109812207969377622615440752640, 53397692462483465346961668429307836, 1775866125092261344436828225211633500160, 75857512919848315654302238627976991244564300

Offset: 0

Vaclav Kotesovec, May 15 2025

nonn

Cf. A032443, A345876, A209289/2, A383853, A383917.

Join[{1}, Table[Sum[Binomial[2*n, n-k]*k^(3*n), {k, 0, n}], {n, 1, 15}]]

a(n) ~ 2^(2*n + 1/2) * r^(3*n + 1) * n^(3*n) / (sqrt(3 - r^2) * exp(3*n) * (1 - r^2)^n), where r = 0.92488761106894648930384927930334708844525256369797556858640... is the root of the equation (1 + r)/(1 - r) = exp(3/r).

A383917 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(5*n).

Original entry on oeis.org

1, 1, 1028, 14545530, 1127435263168, 309320354959336350, 232325928732003715014144, 403150958104730561230009068564, 1432706082674749593552098155989352448, 9528431104471630510834164178027409070527670, 110580781643902847320855308323644986008860441968640

Offset: 0

Vaclav Kotesovec, May 15 2025

nonn

In general, for m>=1, Sum_{k=0..n} binomial(2*n, n-k) * k^(m*n) ~ 2^(2*n + 1/2) * r^(m*n + 1) * n^(m*n) / (sqrt(m + (2-m)*r^2) * exp(m*n) * (1 - r^2)^n), where r is the root of the equation (1 + r)/(1 - r) = exp(m/r).

Cf. A032443 (m=0), A345876 (m=1), A209289/2 (m=2), A383916 (m=3), A383853 (m=4).

Join[{1}, Table[Sum[Binomial[2*n, n-k]*k^(5*n), {k, 0, n}], {n, 1, 12}]]

a(n) ~ 2^(2*n + 1/2) * r^(5*n + 1) * n^(5*n) / (sqrt(5 - 3*r^2) * exp(5*n) * (1 - r^2)^n), where r = 0.98743428968604456152277643726278132237092161504496484119319... is the root of the equation (1 + r)/(1 - r) = exp(5/r).

cp's OEIS Frontend

A383853 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(4*n).

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A383916 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(3*n).

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Programs

Mathematica

Formula

A383917 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(5*n).

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Author

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Mathematica

Formula

cp's OEIS Frontend

A383853 a(n) = Sum_{k=0..n} binomial(2*n, k) * (n-k)^(4*n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A383916 a(n) = Sum_{k=0..n} binomial(2*n, k) * (n-k)^(3*n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A383917 a(n) = Sum_{k=0..n} binomial(2*n, k) * (n-k)^(5*n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A383853 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(4*n).

A383916 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(3*n).

A383917 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(5*n).