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 A007446 Exponentiation of e.g.f. for primes. (Formerly M1785) 9
 1, 2, 7, 31, 162, 973, 6539, 48410, 390097, 3389877, 31534538, 312151125, 3271508959, 36149187780, 419604275375, 5100408982825, 64743452239424, 856157851884881, 11768914560546973, 167841252874889898, 2479014206472819045, 37860543940437797897 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From Tilman Neumann, Oct 05 2008: (Start) a(n) is also given by - substituting the primes (A000040) into (the simplest) Faa di Bruno's formula, or - the complete Bell polynomial of the first n prime arguments, or - computing n-th moments from the first n primes as cumulants 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) REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..500 FORMULA E.g.f.: exp(Sum_{k>=1} prime(k)*x^k/k!). - Ilya Gutkovskiy, Nov 26 2017 EXAMPLE From Tilman Neumann, Oct 05 2008: (Start) Let p_i denote the i-th prime A000040(i). Then a(1)=2 = 1*p_1 a(2)=7 = 1*p_2 + 1*p_1^2 a(3)=31 = 1*p_3 + 3*p_2*p_1 + 1*p_1^3 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 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 (End) MAPLE a:= proc(n) option remember; `if`(n=0, 1,       add(binomial(n-1, j-1)*ithprime(j)*a(n-j), j=1..n))     end: seq(a(n), n=0..30);  # Alois P. Heinz, Mar 18 2015 MATHEMATICA 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 *) Table[Sum[BellY[n, k, Prime[Range[n]]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *) PROG (MuPAD) completeBellMatrix := proc(x, n) // x - vector x[1]...x[m], m>=n local i, j, M; begin M:=matrix(n, n): // zero-initialized for i from 1 to n-1 do M[i, i+1]:=-1: end_for: for i from 1 to n do for j from 1 to i do M[i, j] := binomial(i-1, j-1)*x[i-j+1]: end_for: end_for: return (M): end_proc: completeBellPoly := proc(x, n) begin return (linalg::det(completeBellMatrix(x, n))): end_proc: x:=[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]: for i from 1 to 10 do print(i, completeBellPoly(x, i)): end_for: // Tilman Neumann, Oct 05 2008 CROSSREFS Cf. A036040, A080575. - Tilman Neumann, Oct 05 2008 Sequence in context: A030966 A009132 A125275 * A277396 A227119 A002872 Adjacent sequences:  A007443 A007444 A007445 * A007447 A007448 A007449 KEYWORD easy,nonn AUTHOR STATUS approved

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Last modified May 28 14:47 EDT 2020. Contains 334684 sequences. (Running on oeis4.)