A384524 Expansion of e.g.f. 2/(7 - 5*exp(2*x)).

1, 5, 60, 1070, 25440, 756080, 26964960, 1121963120, 53351831040, 2854122433280, 169649803023360, 11092432778385920, 791204615734640640, 61138238969353748480, 5087702653663698677760, 453621615686933964830720, 43141424825262182799114240, 4359374368561019960377671680

Offset: 0

Author

Seiichi Manyama, Jun 01 2025

Keywords

• nonn

Links

• Wikipedia, Polylogarithm.

Crossrefs

Cf. A094419, A384514, A384521, A384522, A384523.

Programs

Maple

A384524:=proc(n)
    add(5^k * 2^(n-k) * k! * combinat[stirling2](n,k) ,k=0..n) ;
end proc:
seq(A384524(n), n=0..40); # R. J. Mathar, Jun 04 2025

PARI

a(n) = (-2)^(n+1)*polylog(-n, 7/5)/7;

Formula

a(n) = (-2)^(n+1)/7 * Li_{-n}(7/5), where Li_{n}(x) is the polylogarithm function.

a(n) = 2^(n+1)/7 * Sum_{k>=0} k^n * (5/7)^k.

a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * k! * Stirling2(n,k).

a(n) = (5/7) * Sum_{k=0..n} 7^k * (-2)^(n-k) * k! * Stirling2(n,k) for n > 0.

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

a(0) = 1; a(n) = 5 * a(n-1) + 7 * Sum_{k=1..n-1} (-2)^(k-1) * binomial(n-1,k) * a(n-k).