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

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

Offset: 0

Author

Vaclav Kotesovec, May 15 2025

Keywords

• nonn

Comments

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).

Crossrefs

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

Cf. A383917.

Programs

Mathematica

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