A305881 Expansion of Product_{k>=1} 1/(1 + prime(k)*x^k).

1, -2, 1, -7, 16, -28, 62, -118, 303, -630, 1152, -2426, 5315, -10718, 20482, -43449, 91111, -179254, 358910, -727829, 1484601, -2995681, 5924606, -11935441, 24382120, -48702245, 96682698, -195063604, 392983826, -784903199, 1569490057, -3146479152, 6317124649, -12652202092

Offset: 0

Author

Ilya Gutkovskiy, Jun 13 2018

Keywords

• sign

Comments

Convolution inverse of A147655.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3321

Crossrefs

Cf. A000040, A002099, A061151, A145519, A147655, A298160, A304791, A305882.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*ithprime(i))))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i$2)*a(i$2), i=0..n-1))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 13 2018

Mathematica

nmax = 33; CoefficientList[Series[Product[1/(1 + Prime[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[Exp[Sum[Sum[(-1)^k Prime[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (-Prime[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*prime(j)^k*x^(j*k)/k).