%I #8 Mar 15 2018 18:14:34

%S 1,-2,1,5,4,-53,-177,282,5759,20355,-83420,-1420133,-6245485,29035652,

%T 648899541,4034393367,-10488623858,-464971765297,-4310935438663,

%U -3489419105786,446500913437911,6423072226704027,30987397708208720,-462727554963927783,-11862200720684515159

%N Expansion of e.g.f. exp(-Sum_{k>=1} prime(k)*x^k/k!).

%H Alois P. Heinz, <a href="/A300661/b300661.txt">Table of n, a(n) for n = 0..542</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F E.g.f.: exp(-Sum_{k>=1} A000040(k)*x^k/k!).

%e E.g.f.: A(x) = 1 - 2*x/1! + x^2/2! + 5*x^3/3! + 4*x^4/4! - 53*x^5/5! - 177*x^6/6! + 282*x^7/7! + ...

%p a:= proc(n) option remember; `if`(n=0, 1, -add(a(n-j)*

%p ithprime(j)*binomial(n-1, j-1), j=1..n))

%p end:

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Mar 10 2018

%t nmax = 24; CoefficientList[Series[Exp[-Sum[Prime[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

%t a[n_] := a[n] = Sum[-Prime[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}]

%Y Cf. A000040, A007446, A007447, A030017, A030018, A300632.

%K sign

%O 0,2

%A _Ilya Gutkovskiy_, Mar 10 2018