A005445 - OEIS

%I M4487 #33 Jan 05 2025 19:51:33

%S 0,1,1,8,16,224,608,13320,41760,1366152,4440312,215100192,655723440,

%T 48242081328,121651212720,14627299801728,24367884018048,

%U 5768946415383552,2780730890516736,2872938805170308352,-2941729703083507968,1764460446550873413120

%N From a Fibonacci-like differential equation.

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

%H Seiichi Manyama, <a href="/A005445/b005445.txt">Table of n, a(n) for n = 0..434</a>

%H P. R. J. Asveld & N. J. A. Sloane, <a href="/A005442/a005442.pdf">Correspondence, 1987</a>

%H P. R. J. Asveld, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/27-4/asveld.pdf">Fibonacci-like differential equations with a polynomial nonhomogeneous term</a>, Fib. Quart. 27 (1989), 303-309.

%F From _Vladeta Jovovic_, Sep 29 2003: (Start)

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

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

%F 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

%t 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 *)

%o (PARI) a(n) = sum(k=0, n, k!*fibonacci(k)*stirling(n, k, 1)); \\ _Michel Marcus_, Oct 30 2015

%o (Magma) [(&+[Factorial(j)*Fibonacci(j)*StirlingFirst(n,j): j in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Nov 21 2022

%o (SageMath)

%o def A005445(n): return sum((-1)^(n+k)*factorial(k)*fibonacci(k)* stirling_number1(n,k) for k in range(n+1))

%o [A005445(n) for n in range(31)] # _G. C. Greubel_, Nov 21 2022

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

%K sign,changed

%O 0,4

%A _Simon Plouffe_

%E More terms from _Vladeta Jovovic_, Sep 29 2003