A007550 Natural numbers exponentiated twice.

1, 4, 20, 127, 967, 8549, 85829, 962308, 11895252, 160475855, 2343491207, 36795832297, 617662302441, 11031160457672, 208736299803440, 4169680371133507, 87648971646028515, 1933298000313801349, 44633323736412392093, 1076069422794010119112

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

The subsequence of primes (for n = 4, 5, 7) begins: 127, 967, 85829. The subsequence of semiprimes (for n = 2, 6) begins: 4, 8549. - Jonathan Vos Post, Feb 09 2011

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 = 1..200
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 166
• N. J. A. Sloane, Transforms

Programs

Maple

exptr:= proc(p) local g; g:= proc(n) option remember; p(n) +add(binomial(n-1, k-1) *p(k) *g(n-k), k=1..n-1) end: end: a:= exptr(exptr(n->n)): seq(a(n), n=1..30); # Alois P. Heinz, Oct 07 2008

Mathematica

a[n_] := Sum[k^(n-k)*Binomial[n, k]*BellB[k], {k, 0, n}]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Feb 11 2014, after Olivier Gérard *)

Formula

E.g.f.: exp(G(x) - 1) - 1, where G(x) = exp(x*exp(x)) = e.g.f. for A000248; clarified by Ilya Gutkovskiy, Jun 25 2018

a(n) = sum( k^(n - k) binomial(n,k) bell(k), k = 0..n ). - Olivier Gérard, Oct 24 2007