A374917 Inverse of the Fibonacci sequence beginning 1,1 with respect to binomial convolution.

1, -1, 0, 3, -5, -18, 113, 35, -3044, 9755, 87999, -882894, -1155935, 69780087, -292042360, -5040306157, 64613044147, 197030202470, -10570955773551, 48865639709115, 1470783141900676, -21819085085811861, -123330624543827305, 6244177033369108298, -28216305335425392575, -1453926618188019546193

Offset: 0

Author

Fernando Miranda, Maria Irene Falcao and Goncalo Carvalho, Jul 23 2024

Keywords

• sign

Comments

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

Links

• J. A. Adell and A. Lekuona, Binomial convolution and transformations of Appell polynomials, J. Math. Anal. Appl. 456(1), pp. 16-33, 2017.
• P. Appell, Sur une Classe de Polynômes, Ann. Sci. École Norm. Sup. 9(2), pp. 119-144, 1880.

Crossrefs

Cf. A000045, A001622, A094436.

Programs

Maple

p:=(1-sqrt(5))/2: q:=(1+sqrt(5))/2:
egf := (1-2*q)/(p*exp(p*x)-q*exp(q*x)): ser := series(egf, x, 27):
seq(n!*simplify(coeff(ser, x, n)), n=0..25); # Peter Luschny, Aug 05 2024

Mathematica

a[0] = 1; a[n_]:=a[n]= -Sum[Binomial[n, k] Fibonacci[k + 1] a[n - k], {k, 1, n}]
(* or, to generate the list L of the first n terms *)
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}]

Formula

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

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

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.