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A030017 a(1) = 1, a(n+1) = Sum_{k = 1..n} p(k)*a(n+1-k), where p(k) is the k-th prime.

%I A030017 #22 Feb 10 2021 14:13:28
%S A030017 1,2,7,25,88,311,1095,3858,13591,47881,168688,594289,2093693,7376120,
%T A030017 25986209,91549913,322532092,1136286727,4003159847,14103208628,
%U A030017 49685873471,175044281583,616684348614,2172590743211,7654078700221,26965465508072,94999850216565
%N A030017 a(1) = 1, a(n+1) = Sum_{k = 1..n} p(k)*a(n+1-k), where p(k) is the k-th prime.
%C A030017 Apply "INVERT" transform to primes.
%H A030017 Alois P. Heinz, <a href="/A030017/b030017.txt">Table of n, a(n) for n = 1..1000</a> (first 200 terms from T. D. Noe)
%H A030017 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A030017 INVERT: a's from b's in 1+Sum a_i x^i = 1/(1-Sum b_i x^i).
%F A030017 G.f: (1-b(x)/(b(x)-1))*x, where b(x) is the g.f. of A000040. - _Mario C. Enriquez_, Dec 13 2016
%e A030017 a(5) = 25*2 +7*3 +2*5 + 1*7 = 88.
%p A030017 a:= proc(n) option remember; `if`(n=1, 1,
%p A030017       add(a(n-i)*ithprime(i), i=1..n-1))
%p A030017     end:
%p A030017 seq(a(n), n=1..29);  # _Alois P. Heinz_, Feb 10 2021
%t A030017 CoefficientList[ Series[ 1/(1 - Sum[ Prime[ n ]*x^n, {n, 1, 25} ] ), {x, 0, 25} ], x ]
%t A030017 (* Second program: *)
%t A030017 a[1] = 1; a[m_] := a[m] = Sum[Prime@ k  a[m - k], {k, m - 1}]; Table[a@ n, {n, 25}] (* _Michael De Vlieger_, Dec 13 2016 *)
%Y A030017 Row sums of A340991(n-1).
%Y A030017 Cf. A000040.
%K A030017 easy,nonn,nice
%O A030017 1,2
%A A030017 _N. J. A. Sloane_