OFFSET
1,3
COMMENTS
In an increasing rooted graph, nodes are numbered and the numbers increase as you move away from the root.
(a(n+1)/a(n))/n tends to 1/A073003 = 1.676875... (same limit as A029768). - Vaclav Kotesovec, Jul 26 2014
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..410 (first 200 terms from Vincenzo Librandi)
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]
FORMULA
Shifts left when exponentiated twice.
Conjecture: a(n) = Sum_{i=0..2^(n-2) - 1} b(i) for n > 1 with a(1) = 1 where b(n) = (L(n) + 2)*b(f(n)) + Sum_{k=0..L(n) - 1} (1 - R(n,k))*b(f(n) + 2^k*(1 - R(n,k))) for n > 0 with b(0) = 1, L(n) = A000523(n), f(n) = A053645(n) and where R(n,k) = floor(n/2^k) mod 2. Here R(n,k) is the (k+1)-th bit from the right side in the binary expansion of n. - Mikhail Kurkov, Jul 21 2024
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(exptr(a)): a:= n-> `if`(n=0, 1, b(n-1)): 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[ exptr[a] ]; a[n_] := If[n == 0, 1, b[n-1]]; Table[ a[n], {n, 1, 19}] (* Jean-François Alcover, May 10 2012, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,eigen,nice
AUTHOR
EXTENSIONS
New description from Christian G. Bower, Oct 15 1998
STATUS
approved