A061396 - OEIS

%I #30 Oct 19 2019 04:16:16

%S 1,1,2,6,20,73,281,1124,4618,19387,82765,358245,1568458,6933765,

%T 30907194,138760603,626898401,2847946941,13001772692,59618918444,

%U 274463781371,1268064807409,5877758070220,27325789128330,127384553264327,595318139942874,2788598203340643,13090395266913748,61571972632103632

%N Number of "rooted index-functional forests" (Riffs) on n nodes. Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.

%D J. Awbrey, personal journal, circa 1978. Letter to _N. J. A. Sloane_, 1980-Aug-04.

%D G. Balzarotti and P. P. Lava, 103 Curiosità Matematiche, Ulrico Hoepli, Milano, Italy, 2010, pp. 269-271.

%H V. Jovovic, <a href="/A061396/b061396.txt">Table of n, a(n) for n=0..100</a>

%H J. Awbrey, <a href="/A061396/a061396a.txt">Illustration of initial terms</a>

%H Jon Awbrey, <a href="/A061396/a061396.pdf">Letter to N. J. A. Sloane, June 1979</a>

%H Jon Awbrey, <a href="/A061396/a061396_1.pdf">Letter to N. J. A. Sloane, August 1980</a>

%H J. Awbrey, <a href="https://oeis.org/wiki/Riffs_and_Rotes">Riffs and Rotes</a>

%H V. Jovovic, <a href="/A061396/a061396.txt">First 100 terms</a>

%F 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.

%e 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:

%e 2 = (prime_1)^1 = (p_1)^1, briefly, p, weight of 1 node => a(1) = 1.

%e 3 = (prime_2)^1 = (p_2)^1, briefly, p_p, weight of 2 nodes and

%e 4 = (prime_1)^2 = (p_1)^2, briefly, p^p, weight of 2 nodes => a(2) = 2.

%p 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:

%t m = 30; a[0] = 1;

%t 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}];

%t a /@ Range[0, m] (* _Jean-François Alcover_, Oct 19 2019 *)

%Y Cf. A062504, A062860.

%K nice,nonn,easy

%O 0,3

%A _Jon Awbrey_, Jun 09 2001

%E Corrected and extended with Maple program by _Vladeta Jovovic_ and _David W. Wilson_, Jun 20 2001