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A030016 Inverse binomial transform of {1, primes}.

%I A030016 #51 Jul 08 2025 19:27:10
%S A030016 1,1,0,1,-2,5,-14,37,-90,205,-442,899,-1700,2913,-4302,4747,-1080,
%T A030016 -14001,55336,-150395,346164,-716967,1369430,-2432789,4002994,
%U A030016 -5964749,7525018,-6123027,-4900092,40900519,-134308944,348584679,-798958750
%N A030016 Inverse binomial transform of {1, primes}.
%H A030016 Alois P. Heinz, <a href="/A030016/b030016.txt">Table of n, a(n) for n = 0..1000</a>
%H A030016 Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%H A030016 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A030016 a(n) = A008578(n+1) - sum(k=1..n, C(n,k-1)*a(k-1)). - _Vladimir Kruchinin_, Mar 05 2011
%p A030016 b:= n-> `if`(n=1, 1, ithprime(n-1)):
%p A030016 a:= proc(n) option remember;
%p A030016        b(n+1) -add(binomial(n, k-1) *a(k-1), k=1..n)
%p A030016     end:
%p A030016 seq(a(n), n=0..35);  # _Alois P. Heinz_, Mar 06 2011
%t A030016 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_ *)
%K A030016 sign
%O A030016 0,5
%A A030016 _N. J. A. Sloane_