OFFSET
0,4
COMMENTS
Coefficient of the e^(2x) term in the numerator of the n-th derivative of 1/(2-e^x).
This sequence, multiplied by 8, appears in a combinatorial problem about DNA chips. - Bruno Petazzoni (bruno(AT)enix.org), Apr 18 2007
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-20,16).
FORMULA
From Paul Barry, Jan 28 2005: (Start)
G.f.: x^2/((1-2*x)^2*(1-4*x)).
a(n) = Sum_{k=0..n} (-1)^k*3^(n-k)*binomial(n, k)*floor(k/2). (End)
a(n) = 4^(n-1) - (n+1)*2^(n-2). - Bruno Petazzoni (bruno(AT)enix.org), Apr 18 2007
a(n+1) = Sum_{k=0..n} k*2^(2*n-1-k). - Philippe Deléham , Oct 29 2013
E.g.f.: (1/4)*(exp(4*x) - (1 + 2*x)*exp(2*x)). - G. C. Greubel, Mar 21 2022
EXAMPLE
a(2)=1 because there are two maxima in 2,0,1 and 1,0,2
MATHEMATICA
d = Drop[ Flatten[ CoefficientList[ Table[ Simplify[ D[1/(2 - E^x), {x, n}]*(E^x - 2)^(n + 1)/E^x], {n, 2, 24}], E^x]], 1]; a = {}; Do[AppendTo[a, Abs[d[[n(n + 1)/2]]]], {n, 23}]; a (* Robert G. Wilson v, Dec 01 2004 *)
LinearRecurrence[{8, -20, 16}, {0, 0, 1}, 30] (* Harvey P. Dale, Apr 21 2020 *)
PROG
(Magma) [4^(n-1)-(n+1)*2^(n-2): n in [0..30]]; // Vincenzo Librandi, Jul 18 2019
(Sage) [2^(n-2)*(2^n -(n+1)) for n in (0..30)] # G. C. Greubel, Mar 21 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Anthony C Robin, Nov 29 2004
EXTENSIONS
Edited by Robert G. Wilson v, Dec 01 2004
Definition corrected by Bruno Petazzoni (bruno(AT)enix.org), Apr 13 2007
New and simpler definition from R. H. Hardin, Aug 09 2007
STATUS
approved