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%I #31 Jul 10 2021 07:24:24
%S 1,2,8,42,360,4800,102960,3538080,196035840,17520703200,2529842515200,
%T 590412901478400,222813349683724800,136001024583142118400,
%U 134285149587387262464000,214504624277084224347264000,554361997358383529330695680000
%N (Sum of first n Fibonacci numbers) times (product of first n Fibonacci numbers).
%C Let F(1), F(2), F(3), ... be the Fibonacci numbers 1, 1, 2, .... For k=1, we define the tree T(1) the path on two vertices with one identified as the root r. We assign the edge-weight F(1). T(2) is obtained from T(1) by attaching F(2) vertex to the pendents in T(1) except r. In T(2), r is retained as in T(1) and the new edge-weight is assigned as F(2). For k>1, T(k) is obtained from T(k-1) by attaching F(k) vertices to pendents in T(k-1) except r. In T(k), r is retained as in T(k-1) and all the new edge-weights are assigned F(k). With D(1)=1, for k>1 let D(k)=Sum of all distances d(r,x) taken across all vertices x in T(k). By induction it follows that for k>1, D(k)-D(k-1) is this sequence.
%C Retaining the notation of D(k) above, it follows, for k>1, that if D(k)=a(1)F(1)+ - - - +a(k)F(k) then D(k+1)=b(1)F(1)+ - - - +b(k)F(k)+b(k+1)F(k+1) where b(k+1) is the number of leaf nodes in T(k+1).
%H Charles R Greathouse IV, <a href="/A191994/b191994.txt">Table of n, a(n) for n = 1..97</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciFactorialConstant.html">Fibonacci Factorial Constant</a>
%F a(n) ~ C*sqrt(phi^(n^2 + 3*n + 4)/5^(n+1)) where C = A062073 and phi = (1+sqrt(5))/2.
%F a(n) = (F(n+2)-1) * Product_{k=1..n} F(k). - _Franklin T. Adams-Watters_, Jun 23 2011
%o (PARI) s=0;p=1;for(n=1,40,f=fibonacci(n);s+=f;p*=f;print1(s*p", ")) \\ _Charles R Greathouse IV_, Jun 21 2011
%o (PARI) a(n)=prod(k=1,n,fibonacci(k))*(fibonacci(n+2)-1) /* _Franklin T. Adams-Watters_, Jun 23 2011 */
%Y Cf. A000071 (sum of Fibonacci numbers), A003266 (product of Fibonacci numbers).
%Y Cf. A062073 (Fibonacci factorial constant).
%K easy,nonn
%O 1,2
%A _K.V.Iyer_, _Venkata Subba Reddy P._, _Charles R Greathouse IV_, Jun 21 2011
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