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A206296 Prime factorization representation of Fibonacci polynomials: a(0) = 1, a(1) = 2, and for n > 1, a(n) = A003961(a(n-1)) * a(n-2).

%I A206296 #54 Jul 06 2025 02:57:17
%S A206296 1,2,3,10,63,2750,842751,85558343750,2098355820117528699,
%T A206296 769999781728184386440152910156250,
%U A206296 2359414683424785920146467280333749864720543920418139851
%N A206296 Prime factorization representation of Fibonacci polynomials: a(0) = 1, a(1) = 2, and for n > 1, a(n) = A003961(a(n-1)) * a(n-2).
%C A206296 These are numbers matched to the Fibonacci polynomials according to the scheme explained in A206284 (see also A104244). In this case, the exponent of the k-th prime p_k in the prime factorization of a(n) indicates the coefficient of term x^(k-1) in the n-th Fibonacci polynomial. See the examples.
%H A206296 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci polynomial</a>
%H A206296 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fibonacci_polynomials">Fibonacci polynomials</a>
%F A206296 From _Antti Karttunen_, Jul 29 2015: (Start)
%F A206296 a(0) = 1, a(1) = 2, and for n >= 2, a(n) = A003961(a(n-1)) * a(n-2).
%F A206296 Other identities. For all n >= 0:
%F A206296 A001222(a(n)) = A000045(n). [When each polynomial is evaluated at x=1.]
%F A206296 A048675(a(n)) = A000129(n). [at x=2.]
%F A206296 A090880(a(n)) = A006190(n). [at x=3.]
%F A206296 (End)
%e A206296 n    a(n)   prime factorization    Fibonacci polynomial
%e A206296 ------------------------------------------------------------
%e A206296 0       1   (empty)                F_0(x) = 0
%e A206296 1       2   p_1                    F_1(x) = 1
%e A206296 2       3   p_2                    F_2(x) = x
%e A206296 3      10   p_3 * p_1              F_3(x) = x^2 + 1
%e A206296 4      63   p_4 * p_2^2            F_4(x) = x^3 + 2x
%e A206296 5    2750   p_5 * p_3^3 * p_1      F_5(x) = x^4 + 3x^2 + 1
%e A206296 6  842751   p_6 * p_4^4 * p_2^3    F_6(x) = x^5 + 4x^3 + 3x
%t A206296 c[n_] := CoefficientList[Fibonacci[n, x], x]
%t A206296 f[n_] := Product[Prime[k]^c[n][[k]], {k, 1, Length[c[n]]}]
%t A206296 Table[f[n], {n, 1, 11}]  (* A206296 *)
%o A206296 (Scheme, with memoization-macro definec)
%o A206296 (definec (A206296 n) (cond ((<= n 1) (+ 1 n)) (else (* (A003961 (A206296 (- n 1))) (A206296 (- n 2))))))
%o A206296 (Python)
%o A206296 from functools import reduce
%o A206296 from sympy import factorint, prime, primepi
%o A206296 from operator import mul
%o A206296 def a003961(n):
%o A206296     F=factorint(n)
%o A206296     return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
%o A206296 l=[1, 2]
%o A206296 for n in range(2, 11):
%o A206296     l.append(a003961(l[n - 1])*l[n - 2])
%o A206296 print(l) # _Indranil Ghosh_, Jun 21 2017
%Y A206296 Cf. A000045, A000129, A001222, A003961, A006190, A049310, A048675, A090880, A104244, A206284.
%Y A206296 Other such mappings:
%Y A206296   polynomial sequence      integer sequence
%Y A206296   -----------------------------------------
%Y A206296   x^n                      A000040
%Y A206296   (x+1)^n                  A007188
%Y A206296   n*x^(n-1)                A062457
%Y A206296   (1-x^n)/(1-x)            A002110
%Y A206296   n + (n-1)x + ... +x^n    A006939
%Y A206296   Stern polynomials        A260443
%K A206296 nonn
%O A206296 0,2
%A A206296 _Clark Kimberling_, Feb 05 2012
%E A206296 a(0) = 1 prepended (to indicate 0-polynomial), Name changed, Comments and Example section rewritten by _Antti Karttunen_, Jul 29 2015