OFFSET
0,3
COMMENTS
If the e.g.f. of F(n) is E(x) and a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*F(k), then the e.g.f. of a(n) is E(x/(1-x))/(1-x).
FORMULA
E.g.f.: (2/sqrt(5))*exp(x/2/(1-x))*sinh(sqrt(5)*(x/2)/(1-x))/(1-x).
a(n) = (4*n - 3)*a(n-1) - 2*(n-1)*(3*n - 5)*a(n-2) + (n-2)^2*(4*n - 7)*a(n-3) - (n-3)^2*(n-2)^2*a(n-4). - Vaclav Kotesovec, Nov 13 2017
a(n) ~ n^(n + 1/4) / (sqrt(10) * phi^(1/4) * exp(n - 2*sqrt(phi*n) + phi/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 13 2017
EXAMPLE
F(n) = 0,1,1,2,3,5,8,13,21,34,55,...
a(3) = C(3,0)^2*3!*F(0) + C(3,1)^2*2!*F(1) + C(3,2)^2*1!*F(2) + C(3,3)^2*0!*F(3) = 1*6*0 + 9*2*1 + 9*1*1 + 1*1*2 = 0 + 18 + 9 + 2 = 29.
MAPLE
b[0]:=0:b[1]:=1:for n from 2 to 30 do b[n]:=b[n-1]+b[n-2] od:
seq(sum('binomial(n, k)^2*(n-k)!*b[k]', 'k'=0..n), n=0..30);
MATHEMATICA
Table[Sum[Binomial[n, k]^2 * (n-k)! * Fibonacci[k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 13 2017 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Miklos Kristof, Apr 25 2005
STATUS
approved