A005445 From a Fibonacci-like differential equation. (Formerly M4487)

0, 1, 1, 8, 16, 224, 608, 13320, 41760, 1366152, 4440312, 215100192, 655723440, 48242081328, 121651212720, 14627299801728, 24367884018048, 5768946415383552, 2780730890516736, 2872938805170308352, -2941729703083507968, 1764460446550873413120

Offset

0,4

References

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..434

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

From Vladeta Jovovic, Sep 29 2003: (Start)

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

E.g.f.: log(1+x)/(1 - log(1+x) - log(1+x)^2). (End)

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

Mathematica

CoefficientList[Series[Log[1+x]/(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)*stirling(n, k, 1)); \\ Michel Marcus, Oct 30 2015
(Magma) [(&+[Factorial(j)*Fibonacci(j)*StirlingFirst(n, j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Nov 21 2022
(SageMath)
def A005445(n): return sum((-1)^(n+k)*factorial(k)*fibonacci(k)* stirling_number1(n, k) for k in range(n+1))
[A005445(n) for n in range(31)] # G. C. Greubel, Nov 21 2022

Crossrefs

Cf. A000045, A000142, A005444, A048994, A320352.

Sequence in context: A061359 A330150 A340586 * A322342 A132377 A276978

Adjacent sequences: A005442 A005443 A005444 * A005446 A005447 A005448

Keyword

sign

Author

Simon Plouffe

Extensions

More terms from Vladeta Jovovic, Sep 29 2003

Status

approved