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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).
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%I #49 Mar 09 2021 19:13:52

%S 1,2,3,10,63,2750,842751,85558343750,2098355820117528699,

%T 769999781728184386440152910156250,

%U 2359414683424785920146467280333749864720543920418139851

%N 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci polynomial</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fibonacci_polynomials">Fibonacci polynomials</a>

%F From _Antti Karttunen_, Jul 29 2015: (Start)

%F a(0) = 1, a(1) = 2, and for n >= 2, a(n) = A003961(a(n-1)) * a(n-2).

%F Other identities. For all n >= 0:

%F A001222(a(n)) = A000045(n). [When each polynomial is evaluated at x=1.]

%F A048675(a(n)) = A000129(n). [at x=2.]

%F A090880(a(n)) = A006190(n). [at x=3.]

%F (End)

%e n a(n) prime factorization Fibonacci polynomial

%e ------------------------------------------------------------

%e 0 1 (empty) F_0(x) = 0

%e 1 2 p_1 F_1(x) = 1

%e 2 3 p_2 F_2(x) = x

%e 3 10 p_3 * p_1 F_3(x) = x^2 + 1

%e 4 63 p_4 * p_2^2 F_4(x) = x^3 + 2x

%e 5 2750 p_5 * p_3^3 * p_1 F_5(x) = x^4 + 3x^2 + 1

%e 6 842751 p_6 * p_4^4 * p_2^3 F_6(x) = x^5 + 4x^3 + 3x

%t c[n_] := CoefficientList[Fibonacci[n, x], x]

%t f[n_] := Product[Prime[k]^c[n][[k]], {k, 1, Length[c[n]]}]

%t Table[f[n], {n, 1, 11}] (* A206296 *)

%o (Scheme, with memoization-macro definec)

%o (definec (A206296 n) (cond ((<= n 1) (+ 1 n)) (else (* (A003961 (A206296 (- n 1))) (A206296 (- n 2))))))

%o (Python)

%o from sympy import factorint, prime, primepi

%o from operator import mul

%o def a003961(n):

%o F=factorint(n)

%o return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])

%o l=[1, 2]

%o for n in range(2, 11):

%o l.append(a003961(l[n - 1])*l[n - 2])

%o print(l) # _Indranil Ghosh_, Jun 21 2017

%Y Cf. A000045, A000129, A001222, A003961, A006190, A049310, A048675, A090880, A104244, A206284.

%Y Other such mappings:

%Y polynomial sequence integer sequence

%Y -----------------------------------------

%Y x^n A000040

%Y (x+1)^n A007188

%Y n*x^(n-1) A062457

%Y (1-x^n)/(1-x) A002110

%Y n + (n-1)x + ... +x^n A006939

%Y Stern polynomials A260443

%K nonn

%O 0,2

%A _Clark Kimberling_, Feb 05 2012

%E a(0) = 1 prepended (to indicate 0-polynomial), Name changed, Comments and Example section rewritten by _Antti Karttunen_, Jul 29 2015