A007548 Shifts 3 places left under exponentiation.

1, 1, 1, 1, 2, 5, 15, 53, 213, 961, 4808, 26405, 157965, 1022573, 7122441, 53118601, 422362118, 3566967917, 31887812715, 300848966213, 2987359924149, 31143724848889, 340113005563268, 3882897830626949, 46254432194746377, 573938743829923349, 7406289665830324689

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• eigen

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]
• Ronald Orozco López, Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients, Universidad de los Andes (Colombia 2021).

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: b:= exptr(a): a:= n-> `if`(n<=2, 1, b(n-3)): seq(a(n), n=1..30); # Alois P. Heinz, Oct 07 2008

Mathematica

Exptr[p_] := Module[{g}, g[n_] := g[n] = p[n] + Sum [Binomial[n-1, k-1]*p[k]*g[n-k], {k, 1, n-1}]; g]; b = Exptr [a]; a[n_] := If[n <= 2, 1, b[n-3]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, May 06 2014, after Alois P. Heinz *)

Formula

E.g.f. A(x) satisfies differential equation A'''(x)=exp(A(x)), A'(0)=1, A''(0)=1, A'''(0)=1. - Vladimir Kruchinin, Nov 19 2011