A100575 Half the number of permutations of 0..n with exactly two maxima.

0, 0, 1, 8, 44, 208, 912, 3840, 15808, 64256, 259328, 1042432, 4180992, 16748544, 67047424, 268304384, 1073463296, 4294377472, 17178624000, 68716855296, 274872401920, 1099500093440, 4398022393856, 17592135712768, 70368639320064

Offset: 0

Author

Anthony C Robin, Nov 29 2004

Keywords

• nonn

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

Examples

			a(2)=1 because there are two maxima in 2,0,1 and 1,0,2
		

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).

Crossrefs

Cf. A000431.

Programs

Magma

[4^(n-1)-(n+1)*2^(n-2): n in [0..30]]; // Vincenzo Librandi, Jul 18 2019

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 *)

Sage

[2^(n-2)*(2^n -(n+1)) for n in (0..30)] # G. C. Greubel, Mar 21 2022

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

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