A061396 Number of "rooted index-functional forests" (Riffs) on n nodes. Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.
1, 1, 2, 6, 20, 73, 281, 1124, 4618, 19387, 82765, 358245, 1568458, 6933765, 30907194, 138760603, 626898401, 2847946941, 13001772692, 59618918444, 274463781371, 1268064807409, 5877758070220, 27325789128330, 127384553264327, 595318139942874, 2788598203340643, 13090395266913748, 61571972632103632
Offset: 0
Examples
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.
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
Programs
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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:
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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 *)
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.
Extensions
Corrected and extended with Maple program by Vladeta Jovovic and David W. Wilson, Jun 20 2001