OFFSET
0,3
REFERENCES
J. Awbrey, personal journal, circa 1978. Letter to N. J. A. Sloane, 1980-Aug-04.
G. Balzarotti and P. P. Lava, 103 Curiosità Matematiche, Ulrico Hoepli, Milano, Italy, 2010, pp. 269-271.
LINKS
V. Jovovic, Table of n, a(n) for n=0..100
J. Awbrey, Illustration of initial terms
Jon Awbrey, Letter to N. J. A. Sloane, June 1979
Jon Awbrey, Letter to N. J. A. Sloane, August 1980
J. Awbrey, Riffs and Rotes
V. Jovovic, First 100 terms
FORMULA
G.f. A(x) = 1 + x + 2*x^2 + 6*x^3 + ... satisfies A(x) = Product_{j >= 0} (1 + x^(j+1)*A(x))^a_j.
EXAMPLE
These structures come from recursive primes' factorizations of natural numbers, where the recursion proceeds on both the exponents (^k) and the indices (_k) of the primes invoked in the factorization:
2 = (prime_1)^1 = (p_1)^1, briefly, p, weight of 1 node => a(1) = 1.
3 = (prime_2)^1 = (p_2)^1, briefly, p_p, weight of 2 nodes and
4 = (prime_1)^2 = (p_1)^2, briefly, p^p, weight of 2 nodes => a(2) = 2.
MAPLE
a(0) := 1: for k from 1 to 30 do A := add(a(i)*x^i, i=0..k): B := mul((1+x^(j+1)*A)^a(j), j=0..k-1): a(k) := coeff(series(B, x, k+1), x, k): printf(`%d, `, a(k)); od:
MATHEMATICA
m = 30; a[0] = 1;
Do[A[x_] = Product[(1+x^(j+1)*Sum[a[i]*x^i, {i, 0, k}])^a[j], {j, 0, k-1}]; a[k] = SeriesCoefficient[A[x], {x, 0, k}], {k, 1, m}];
a /@ Range[0, m] (* Jean-François Alcover, Oct 19 2019 *)
CROSSREFS
KEYWORD
nice,nonn,easy
AUTHOR
Jon Awbrey, Jun 09 2001
EXTENSIONS
Corrected and extended with Maple program by Vladeta Jovovic and David W. Wilson, Jun 20 2001
STATUS
approved