This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A320565 #11 Feb 16 2025 08:33:56
%S A320565 0,1,1,10,33,701,4033,132301,1089921,48460114,520210801,29215223489,
%T A320565 386721507745,26250621340841,413242502386337,32899021525375426,
%U A320565 600383148312628737,54846079150716441949,1138470675779123657425,117372939125452004885621
%N A320565 a(n) = ((1 + sqrt(4*n^2 + 1))^n - (1 - sqrt(4*n^2 + 1))^n)/(2^n * sqrt(4*n^2 + 1)).
%C A320565 a(0) = 0 assuming 0^0 = 1, or using the limit for n -> 0 (assuming n is a real variable); the same value for a(0) arises from other formulae for this sequence.
%H A320565 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci Polynomial</a>
%H A320565 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fibonacci_polynomials">Fibonacci polynomials</a>
%F A320565 a(n) = 2^(1 - n) * Sum_{k=0..floor(n/2)} binomial(n, 2*k + 1)*(4*n^2 + 1)^k.
%F A320565 a(n) = 2 * (i*n)^n * sinh(n*arctanh(sqrt(4*n^2 + 1)))/sqrt(4*n^2 + 1), assuming 0^0 = 1 for n = 0.
%F A320565 For n > 0, a(n) = n^(n - 1) * F_n(1/n), where F_n(x) is the Fibonacci polynomial.
%F A320565 For n > 0, a(n) = sqrt(Pi/4)*i*(-i*n)^n*LegendreP((n - 1)/2, -1/2, -1/(2*n^2) - 1) / (4*n^2 + 1)^(1/4). - _Peter Luschny_, Oct 15 2018
%t A320565 a[0] = Limit[n^(n - 1) Fibonacci[n, 1/n], n -> 0]; (* a[0] = 0 *)
%t A320565 a[n_] := a[n] = n^(n - 1) Fibonacci[n, 1/n];
%t A320565 Table[a[n], {n, 0, 19}]
%o A320565 (PARI) for(n=0,20, print1( 2^(1-n)*sum(k=0,floor(n/2), binomial(n,2*k+1)*(4*n^2+1)^k) , ", ")) \\ _G. C. Greubel_, Oct 15 2018
%o A320565 (Magma) [2^(1-n)*(&+[Binomial(n,2*k+1)*(4*n^2+1)^k: k in [0..Floor(n/2)]]): n in [0..20]]; // _G. C. Greubel_, Oct 15 2018
%Y A320565 Cf. A084844, A320519, A320534.
%K A320565 nonn
%O A320565 0,4
%A A320565 _Vladimir Reshetnikov_, Oct 15 2018