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

1,3

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).

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A182716 A143281 A098057 * A074029 A248729 A138653

Adjacent sequences: A289689 A289690 A289691 * A289693 A289694 A289695

Keyword

nonn,easy

Author

R. J. Mathar, Jul 09 2017

Status

approved