A005444 From a Fibonacci-like differential equation.

1, 1, 3, 8, 50, 214, 2086, 11976, 162816, 1143576, 20472504, 165910128, 3785092032, 33908109936, 967508478192, 9252123203712, 327062428940160, 3236057604910080, 141403289873955840, 1404243298160352000, 76168955916831029760, 735206146073008508160

Offset: 0

Author

Simon Plouffe, N. J. A. Sloane

Keywords

• sign
• easy

Comments

Sequence is signed: first negative term is a(35) = -230450728485788167742674544892530875760640. - Vladeta Jovovic, Sep 29 2003

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Georg Fischer, Table of n, a(n) for n = 0..100
• P. R. J. Asveld & N. J. A. Sloane, Correspondence, 1987
• P. R. J. Asveld, Fibonacci-like differential equations with a polynomial nonhomogeneous term, Fib. Quart. 27 (1989), 303-309.

Crossrefs

Cf. A000045, A000142, A005445, A048994.

Programs

Magma

[(&+[Factorial(j)*Fibonacci(j+1)*StirlingFirst(n,j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Nov 21 2022

Mathematica

CoefficientList[Series[1/(1-Log[1+x]-(Log[1+x])^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

PARI

a(n) = sum(k=0, n, k!*fibonacci(k+1)*stirling(n, k, 1)); \\ Michel Marcus, Oct 30 2015

SageMath

def A005444(n): return sum((-1)^(n+k)*factorial(k)*fibonacci(k+1)* stirling_number1(n,k) for k in (0..n))
[A005444(n) for n in range(31)] # G. C. Greubel, Nov 21 2022

Formula

a(n) = Sum_{k=0..n} k!*Fibonacci(k+1)*Stirling1(n, k).

E.g.f.: 1/(1 - log(1+x) - log(1+x)^2). - Vladeta Jovovic, Sep 29 2003

a(n) ~ n! * (-1)^n * exp(n*(1+sqrt(5))/2) / (sqrt(5)*(exp((1+sqrt(5))/2)-1)^(n+1)). - Vaclav Kotesovec, Oct 01 2013