A289692 The number of partitions of [n] with exactly 2 blocks without peaks.

0, 1, 2, 4, 8, 15, 27, 48, 85, 150, 264, 464, 815, 1431, 2512, 4409, 7738, 13580, 23832, 41823, 73395, 128800, 226029, 396654, 696080, 1221536, 2143647, 3761839, 6601568, 11584945

Offset: 1

Author

R. J. Mathar, Jul 09 2017

Keywords

• nonn
• easy

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..300
• T. Mansour and M. Shattuck, Counting Peaks and Valleys in a Partition of a Set, J. Int. Seq. 13 (2010), 10.6.8, Table 1.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-1).

Crossrefs

Cf. A005251 (first differences), A289693 (3 blocks), A289694 (4 blocks).

Programs

GAP

a:=[0, 1, 2, 4]; for n in [5..10^2] do a[n]:=3*a[n-1]-3*a[n-2]+2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Jan 25 2018

Magma

I:=[0, 1, 2, 4]; [n le 4 select I[n] else 3*Self(n-1)-3*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jan 26 2018

Maple

a := proc(n) option remember: if n = 1 then 0 elif n = 2 then 1 elif n=3 then 2 elif n=4 then 4 elif  n >= 5 then 3*procname(n-1) -3*procname(n-2)+2*procname(n-3)-procname(n-4) fi; end:
seq(a(n), n = 0..100); # Muniru A Asiru, Jan 25 2018

Mathematica

LinearRecurrence[{3, -3, 2, -1}, {0, 1, 2, 4}, 40] (* Vincenzo Librandi, Jan 26 2018 *)

Formula

From Colin Barker, Nov 07 2017: (Start)

G.f.: x^2*(1 - x + x^2) / ((1 - x)*(1 - 2*x + x^2 - x^3)).

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - a(n-4) for n>4. (End)

a(n) = A077855(n-2) - A005314(n-2) for n>1. - John Molokach, Jan 23 2018

a(n) - a(n-1) = A005251(n). - R. J. Mathar, Mar 11 2021