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

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

Offset: 0

Author

Vaclav Kotesovec, May 15 2025

Keywords

• nonn

Comments

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).

Crossrefs

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

Programs

Mathematica

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

Formula

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).