A384525 - OEIS

%I #12 Jun 01 2025 09:57:40

%S 1,1,7,61,679,9445,158095,3088765,68958295,1731875605,48328686175,

%T 1483501074925,49677478279975,1802159471217925,70406303657894575,

%U 2947087948180076125,131584088098220272375,6242270620707298139125,313548981075158413477375

%N Expansion of e.g.f. 5/(6 - exp(5*x)).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polylogarithm">Polylogarithm</a>.

%F a(n) = (-5)^(n+1)/6 * Li_{-n}(6), where Li_{n}(x) is the polylogarithm function.

%F a(n) = 5^(n+1) * Sum_{k>=0} k^n * (1/6)^(k+1).

%F a(n) = Sum_{k=0..n} 5^(n-k) * k! * Stirling2(n,k).

%F a(n) = (1/6) * Sum_{k=0..n} 6^k * (-5)^(n-k) * k! * Stirling2(n,k) for n > 0.

%F a(0) = 1; a(n) = Sum_{k=1..n} 5^(k-1) * binomial(n,k) * a(n-k).

%F a(0) = 1; a(n) = a(n-1) + 6 * Sum_{k=1..n-1} (-5)^(k-1) * binomial(n-1,k) * a(n-k).

%o (PARI) a(n) = (-5)^(n+1)*polylog(-n, 6)/6;

%Y Cf. A326323.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jun 01 2025