|
%I M2766 #32 Nov 21 2022 03:02:19
%S 1,1,3,8,50,214,2086,11976,162816,1143576,20472504,165910128,
%T 3785092032,33908109936,967508478192,9252123203712,327062428940160,
%U 3236057604910080,141403289873955840,1404243298160352000,76168955916831029760,735206146073008508160
%N From a Fibonacci-like differential equation.
%C Sequence is signed: first negative term is a(35) = -230450728485788167742674544892530875760640. - _Vladeta Jovovic_, Sep 29 2003
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Georg Fischer, <a href="/A005444/b005444.txt">Table of n, a(n) for n = 0..100</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="http://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 a(n) = Sum_{k=0..n} k!*Fibonacci(k+1)*Stirling1(n, k).
%F E.g.f.: 1/(1 - log(1+x) - log(1+x)^2). - _Vladeta Jovovic_, Sep 29 2003
%F 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
%t CoefficientList[Series[1/(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+1)*stirling(n, k, 1)); \\ _Michel Marcus_, Oct 30 2015
%o (Magma) [(&+[Factorial(j)*Fibonacci(j+1)*StirlingFirst(n,j): j in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Nov 21 2022
%o (SageMath)
%o def A005444(n): return sum((-1)^(n+k)*factorial(k)*fibonacci(k+1)* stirling_number1(n,k) for k in (0..n))
%o [A005444(n) for n in range(31)] # _G. C. Greubel_, Nov 21 2022
%Y Cf. A000045, A000142, A005445, A048994.
%K sign,easy
%O 0,3
%A _Simon Plouffe_, _N. J. A. Sloane_
|
