A191994 (Sum of first n Fibonacci numbers) times (product of first n Fibonacci numbers).

1, 2, 8, 42, 360, 4800, 102960, 3538080, 196035840, 17520703200, 2529842515200, 590412901478400, 222813349683724800, 136001024583142118400, 134285149587387262464000, 214504624277084224347264000, 554361997358383529330695680000

Offset

1,2

Comments

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.

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).

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..97

Eric Weisstein's World of Mathematics, Fibonacci Factorial Constant

Formula

a(n) ~ C*sqrt(phi^(n^2 + 3*n + 4)/5^(n+1)) where C = A062073 and phi = (1+sqrt(5))/2.

a(n) = (F(n+2)-1) * Product_{k=1..n} F(k). - Franklin T. Adams-Watters, Jun 23 2011

Prog

(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
(PARI) a(n)=prod(k=1, n, fibonacci(k))*(fibonacci(n+2)-1) /* Franklin T. Adams-Watters, Jun 23 2011 */

Crossrefs

Cf. A000071 (sum of Fibonacci numbers), A003266 (product of Fibonacci numbers).

Cf. A062073 (Fibonacci factorial constant).

Sequence in context: A320343 A002856 A093461 * A153524 A153552 A295199

Adjacent sequences: A191991 A191992 A191993 * A191995 A191996 A191997

Keyword

easy,nonn

Author

K.V.Iyer, Venkata Subba Reddy P., Charles R Greathouse IV, Jun 21 2011

Status

approved