A383917 -id:A383917 - cp's OEIS frontend

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

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).

A383930 a(n) = Sum_{k=0..n} (-1)^k * binomial(2n, k) (n-k)^(5*n).

Original entry on oeis.org

1, 1, 1020, 14152314, 1071646712640, 286802348769420190, 209974096349134108992000, 355016116241074708829385321492, 1228958111984894631846657261766656000, 7960240318398277162915923478914410838135990, 89961580311571094335785117669395413813764096000000

Offset: 0

Vaclav Kotesovec, May 15 2025

nonn

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

Cf. A002674 (m=2), A383929 (m=3), A298851*A002674 (m=4).

Cf. A383917.

Join[{1}, Table[Sum[(-1)^(n-k)*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) * (r^2 - 1)^n), where r = 1.0145858159274292356581282820876562174881159476120290450838... is the root of the equation (1 + r)/(1 - r) = -exp(5/r).

cp's OEIS Frontend

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

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A383930 a(n) = Sum_{k=0..n} (-1)^k * binomial(2n, k) (n-k)^(5*n).

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cp's OEIS Frontend

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A383930 a(n) = Sum_{k=0..n} (-1)^k * binomial(2*n, k) * (n-k)^(5*n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

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

A383930 a(n) = Sum_{k=0..n} (-1)^k * binomial(2n, k) (n-k)^(5*n).