A384525 Expansion of e.g.f. 5/(6 - exp(5*x)).

1, 1, 7, 61, 679, 9445, 158095, 3088765, 68958295, 1731875605, 48328686175, 1483501074925, 49677478279975, 1802159471217925, 70406303657894575, 2947087948180076125, 131584088098220272375, 6242270620707298139125, 313548981075158413477375

Offset: 0

Author

Seiichi Manyama, Jun 01 2025

Keywords

• nonn

Links

• Wikipedia, Polylogarithm.

Crossrefs

Cf. A326323.

Programs

PARI

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

Formula

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

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

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

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

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

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