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A061396
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Number of "rooted index-functional forests" (Riffs) on n nodes. Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.
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39
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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
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OFFSET
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0,3
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REFERENCES
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MAPLE
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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
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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}];
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CROSSREFS
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KEYWORD
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nice,nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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