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

0,3

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

Cf. A062504, A062860.

Sequence in context: A150139 A052884 A150140 * A230823 A192497 A104632

Adjacent sequences: A061393 A061394 A061395 * A061397 A061398 A061399

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