%I #42 Sep 08 2022 08:45:01
%S 0,1,1,128,2187,78125,2097152,62748517,1801088541,52523350144,
%T 1522435234375,44231334895529,1283918464548864,37281334283719577,
%U 1082404156823183753,31427428360210000000,912473096871571914483
%N Seventh power of Fibonacci numbers A000045.
%C A divisibility sequence; that is, if n divides m, then a(n) divides a(m).
%D D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 1, p. 85, (exercise 1.2.8. Nr. 30) and p. 492 (solution).
%H Vincenzo Librandi, <a href="/A056574/b056574.txt">Table of n, a(n) for n = 0..115</a>
%H Mohammad K. Azarian, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/37-40-2012/azarianIJCMS37-40-2012.pdf">Fibonacci Identities as Binomial Sums</a>, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876. Mathematical Reviews, MR2959001. Zentralblatt MATH, Zbl 1255.05003.
%H Mohammad K. Azarian, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/41-44-2012/azarianIJCMS41-44-2012.pdf">Fibonacci Identities as Binomial Sums II</a>, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059. Mathematical Reviews, MR2980853. Zentralblatt MATH, Zbl 1255.05004.
%H A. Brousseau, <a href="http://www.fq.math.ca/Scanned/6-1/brousseau3.pdf">A sequence of power formulas</a>, Fib. Quart., 6 (1968), 81-83.
%H J. Riordan, <a href="http://dx.doi.org/10.1215/S0012-7094-62-02902-2">Generating functions for powers of Fibonacci numbers</a>, Duke. Math. J. 29 (1962) 5-12.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (21,273,-1092,-1820,1092,273,-21,-1).
%F a(n) = F(n)^7, where F(n) = A000045(n).
%F G.f.: x*p(7, x)/q(7, x) with p(7, x) := sum_{m=0..6} A056588(6, m)*x^m = 1 - 20*x - 166*x^2 + 318*x^3 + 166*x^4 - 20*x^5 - x^6 and q(7, x) := sum_{m=0..8} A055870(8, m)*x^m = (1 + x - x^2)*(1 - 4*x - x^2)*(1 + 11*x - x^2)*(1 - 29*x - x^2) (factorization deduced from Riordan result).
%F Recursion (cf. Knuth's exercise): sum_{m=0..8} A055870(8, m)*a(n-m) = 0, n >= 8; inputs: a(n), n=0..7. a(n) = 21*a(n-1) + 273*a(n-2) - 1092*a(n-3) - 1820*a(n-4) + 1092*a(n-5) + 273*a(n-6) - 21*a(n-7) - a(n-8).
%F a(n+1) = F(n)^7+F(n+1)^7+7*F(n)*F(n+1)*F(n+2)*[2*F(n+1)^2-(-1)^n]^2 = {Sum(0 <= j <= [n/2]; binomial(n-j, j))}^7, for n>=0 (This is Theorem 2.3 (iv) of Azarian's second paper in the references for this sequence). - _Mohammad K. Azarian_, Jun 29 2015
%t f[n_]:=Fibonacci[n]^7; lst={}; Do[AppendTo[lst,f[n]],{n,0,5!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Feb 12 2010 *)
%o (Magma) [Fibonacci(n)^7: n in [0..20]]; // _Vincenzo Librandi_, Jun 04 2011
%o (PARI) a(n)=fibonacci(n)^7 \\ _Charles R Greathouse IV_, Jan 30 2012
%Y Cf. A000045, A007598, A056570, A056571, A056572, A056573, A056588, A055870.
%Y Seventh row of array A103323.
%K nonn,easy
%O 0,4
%A _Wolfdieter Lang_, Jul 10 2000