A380310 Expansion of e.g.f. exp( 1 - 1/(1-5*x)^(1/5) ).

1, -1, -5, -49, -719, -14077, -344909, -10152829, -349045535, -13727327833, -607873987637, -29931556660105, -1622308999459631, -95982568510668373, -6155361624644676989, -425321834949751148053, -31502433469012320013631, -2489898822489054343250737, -209178052238110675644666341

Offset: 0

Author

Seiichi Manyama, Jan 20 2025

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..355

Crossrefs

Cf. A145561, A380308, A380309.

Cf. A000587, A008548, A028575.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(1-1/(1-5*x)^(1/5))))

Formula

a(n) = Sum_{k=0..n} 5^(n-k) * |Stirling1(n,k)| * A000587(k).

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

a(0) = 1; a(n) = -Sum_{k=1..n} A008548(k) * binomial(n-1,k-1) * a(n-k).