A374917 - OEIS

%I #29 Aug 05 2024 17:52:57

%S 1,-1,0,3,-5,-18,113,35,-3044,9755,87999,-882894,-1155935,69780087,

%T -292042360,-5040306157,64613044147,197030202470,-10570955773551,

%U 48865639709115,1470783141900676,-21819085085811861,-123330624543827305,6244177033369108298,-28216305335425392575,-1453926618188019546193

%N Inverse of the Fibonacci sequence beginning 1,1 with respect to binomial convolution.

%C The binomial convolution of this sequence with the Fibonacci sequence beginning 1,1 gives the identity sequence with respect to convolution (A000007).

%H J. A. Adell and A. Lekuona, <a href="https://doi.org/10.1016/j.jmaa.2017.06.077">Binomial convolution and transformations of Appell polynomials</a>, J. Math. Anal. Appl. 456(1),  pp. 16-33, 2017.

%H P. Appell, <a href="https://doi.org/10.24033/asens.186">Sur une Classe de Polynômes</a>, Ann. Sci. École Norm. Sup. 9(2), pp. 119-144, 1880.

%F a(0) = 1, a(n) = -Sum_{k=1..n} binomial(n, k)*a(n - k)*A000045(k+1).

%F E.g.f.: 1/G'(x) where G(x) is the e.g.f. of A000045.

%F The recursion P(0, x) = 1, P(n, x) = x^n - Sum_{k=0..n-1} binomial(n, k)*a(n-k)*P(k, x) defines the so-called Appell-Fibonacci polynomials P(n, x) = Sum_{k=0..n} T(n, k)*x^k, where T(n, k) is the triangular array A094436.

%p p:=(1-sqrt(5))/2: q:=(1+sqrt(5))/2:

%p egf := (1-2*q)/(p*exp(p*x)-q*exp(q*x)): ser := series(egf, x, 27):

%p seq(n!*simplify(coeff(ser, x, n)), n=0..25); # _Peter Luschny_, Aug 05 2024

%t a[0] = 1; a[n_]:=a[n]= -Sum[Binomial[n, k] Fibonacci[k + 1] a[n - k], {k, 1, n}]

%t (* or, to generate the list L of the first n terms *)

%t phi = (1 + Sqrt[5])/2; psi = 1 - phi; L[n_] := CoefficientList[Series[(phi - psi)/(phi Exp[phi x] - psi Exp[psi x]), {x, 0, n}], x] Table[k!, {k, 0, n}]

%Y Cf. A000045, A001622, A094436.

%K sign

%O 0,4

%A _Fernando Miranda_, _Maria Irene Falcao_ and Goncalo Carvalho, Jul 23 2024