A030016 Inverse binomial transform of {1, primes}.

1, 1, 0, 1, -2, 5, -14, 37, -90, 205, -442, 899, -1700, 2913, -4302, 4747, -1080, -14001, 55336, -150395, 346164, -716967, 1369430, -2432789, 4002994, -5964749, 7525018, -6123027, -4900092, 40900519, -134308944, 348584679, -798958750

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
• N. J. A. Sloane, Transforms

Programs

Maple

b:= n-> `if`(n=1, 1, ithprime(n-1)):
a:= proc(n) option remember;
       b(n+1) -add(binomial(n, k-1) *a(k-1), k=1..n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 06 2011

Mathematica

b[n_] := If[n == 1, 1, Prime[n - 1]]; a[n_] := a[n] = b[n + 1] - Sum [Binomial[n, k - 1]*a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)

Formula

a(n) = A008578(n+1) - sum(k=1..n, C(n,k-1)*a(k-1)). - Vladimir Kruchinin, Mar 05 2011