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

0,4

Comments

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

Links

Table of n, a(n) for n=0..25.

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.

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.

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}]

Crossrefs

Cf. A000045, A001622, A094436.

Sequence in context: A022489 A176749 A374649 * A145774 A348778 A270894

Adjacent sequences: A374914 A374915 A374916 * A374918 A374919 A374920

Keyword

sign

Author

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

Status

approved