A330199 Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k)).

1, -1, -2, 1, 1, 98, -39, 3225, 1226, 6459, 12473, 821830, -214739887, -201448561, -8997850614, -514986723363, -1310942141971, -26465356716946, -931753364233567, -1858534483400559, 167210272584038942, -7112146717031426801, 312288595642509829797

Offset: 0

Author

Ilya Gutkovskiy, Dec 05 2019

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A000587, A057625, A209903.

Programs

Mathematica

nmax = 22; CoefficientList[Series[Product[Exp[1 - Exp[x^k]], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n - 1, k - 1] k! DivisorSum[k, 1/#! &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = e.g.f. of complementary Bell numbers (A000587).

E.g.f.: exp(-Sum_{j>=1} Sum_{i>=1} x^(i*j) / i!).

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * A057625(k) * a(n-k).