A005444 From a Fibonacci-like differential equation. (Formerly M2766)

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

Offset

0,3

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.

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

Mathematica

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

Prog

(PARI) a(n) = sum(k=0, n, k!*fibonacci(k+1)*stirling(n, k, 1)); \\ Michel Marcus, Oct 30 2015
(Magma) [(&+[Factorial(j)*Fibonacci(j+1)*StirlingFirst(n, j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Nov 21 2022
(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

Crossrefs

Cf. A000045, A000142, A005445, A048994.

Sequence in context: A306042 A194364 A338439 * A222684 A208801 A019044

Adjacent sequences: A005441 A005442 A005443 * A005445 A005446 A005447

Keyword

sign,easy

Author

Simon Plouffe, N. J. A. Sloane

Status

approved