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A007446 Exponentiation of e.g.f. for primes.

%I A007446 M1785 #34 Nov 27 2017 12:41:00
%S A007446 1,2,7,31,162,973,6539,48410,390097,3389877,31534538,312151125,
%T A007446 3271508959,36149187780,419604275375,5100408982825,64743452239424,
%U A007446 856157851884881,11768914560546973,167841252874889898,2479014206472819045,37860543940437797897
%N A007446 Exponentiation of e.g.f. for primes.
%C A007446 From _Tilman Neumann_, Oct 05 2008: (Start)
%C A007446 a(n) is also given by
%C A007446 - substituting the primes (A000040) into (the simplest) Faa di Bruno's formula, or
%C A007446 - the complete Bell polynomial of the first n prime arguments, or
%C A007446 - computing n-th moments from the first n primes as cumulants
%C A007446 The examples show that the coefficients of the prime power products are just A036040/A080575 (these are just rearrangements of the same coefficients). Moreover, the prime products of the additional terms span the whole space of natural numbers, thus what we see here is a reordering of the natural numbers! (End)
%D A007446 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007446 Alois P. Heinz, <a href="/A007446/b007446.txt">Table of n, a(n) for n = 0..500</a>
%F A007446 E.g.f.: exp(Sum_{k>=1} prime(k)*x^k/k!). - _Ilya Gutkovskiy_, Nov 26 2017
%e A007446 From _Tilman Neumann_, Oct 05 2008: (Start)
%e A007446 Let p_i denote the i-th prime A000040(i). Then
%e A007446 a(1)=2 = 1*p_1
%e A007446 a(2)=7 = 1*p_2 + 1*p_1^2
%e A007446 a(3)=31 = 1*p_3 + 3*p_2*p_1 + 1*p_1^3
%e A007446 a(4)=162= 1*p_4 + 4*p_3*p_1 + 3*p_2^2 + 6*p_2*p_1^2 + 1*p_1^4
%e A007446 a(5)=973= 1*p_5 + 5*p_4*p_1 + 10*p_3*p_2 + 10*p_3*p_1^2 + 15*p_2^2*p_1 + 10*p_2*p_1^3 + 1*p_1^5
%e A007446 (End)
%p A007446 a:= proc(n) option remember; `if`(n=0, 1,
%p A007446       add(binomial(n-1, j-1)*ithprime(j)*a(n-j), j=1..n))
%p A007446     end:
%p A007446 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 18 2015
%t A007446 a[n_] := a[n] = If[n==0, 1, Sum[Binomial[n-1, j-1]*Prime[j]*a[n-j], {j, 1, n}]]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 30 2015, after _Alois P. Heinz_ *)
%t A007446 Table[Sum[BellY[n, k, Prime[Range[n]]], {k, 0, n}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%o A007446 (MuPAD)
%o A007446 completeBellMatrix := proc(x,n)
%o A007446 // x - vector x[1]...x[m], m>=n
%o A007446 local i,j,M;
%o A007446 begin
%o A007446 M:=matrix(n,n): // zero-initialized
%o A007446 for i from 1 to n-1 do
%o A007446 M[i,i+1]:=-1:
%o A007446 end_for:
%o A007446 for i from 1 to n do
%o A007446 for j from 1 to i do
%o A007446 M[i,j] := binomial(i-1,j-1)*x[i-j+1]:
%o A007446 end_for:
%o A007446 end_for:
%o A007446 return (M):
%o A007446 end_proc:
%o A007446 completeBellPoly := proc(x, n)
%o A007446 begin
%o A007446 return (linalg::det(completeBellMatrix(x,n))):
%o A007446 end_proc:
%o A007446 x:=[2,3,5,7,11,13,17,19,23,29]:
%o A007446 for i from 1 to 10 do print(i,completeBellPoly(x,i)): end_for:
%o A007446 // _Tilman Neumann_, Oct 05 2008
%Y A007446 Cf. A036040, A080575. - _Tilman Neumann_, Oct 05 2008
%K A007446 easy,nonn
%O A007446 0,2
%A A007446 _N. J. A. Sloane_