A238159 Number of compositions of n with exactly one part equal to 1 or exactly one part equal to 2.

0, 1, 1, 2, 5, 11, 15, 35, 70, 124, 234, 447, 827, 1529, 2834, 5222, 9587, 17573, 32137, 58641, 106821, 194280, 352824, 639913, 1159238, 2097759, 3792375, 6849778, 12361822, 22292405, 40172089, 72344671, 130203409, 234200988, 421037335, 756538955, 1358728300

Offset: 0

Author

Geoffrey Critzer, Feb 18 2014

Keywords

• nonn

Examples

			a(4) = 5 because we have: 1+3, 3+1, 1+1+2, 1+2+1, 2+1+1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-34,73,-109,137,-147,120,-79,40,3,-20,21,-22,8,-3,1,3,0,1).

Crossrefs

Cf. A006367 exactly one part equal to 1, A079662 exactly one part equal to 2 (with appropriate offset).

Programs

Mathematica

nn=30;a=1/(1-(x/(1-x)-x));b=1/(1-(x/(1-x)-x^2));c=1/(1-(x/(1-x)-x-x^2));CoefficientList[Series[a^2x +b^2x^2-2 c^3x^3,{x,0,nn}],x]
(* or *)
Table[Length[Select[Level[Table[Select[Compositions[n,k],Count[#,0]==0&],{k,1,n}],{2}],Count[#,1]==1||Count[#,2]==1&]],{n,0,10}]

Formula

G.f.: x*A(x)^2 + x^2*B(x)^2 - 2*x^3*C(x)^3 where A(x)=1/(1 - (x/(1-x)-x)), B(x)=1/(1 - (x/(1-x)-x^2)), C(x)=1/(1 - (x/(1-x)-x-x^2)).

a(n) ~ c * n / (2^(n-1) * d^n), where c = 0.02749202171174083217... is the root of the equation -1 + 18*c + 552*c^2 + 4232*c^3 = 0 and d = 0.2849201454990266329... is the root of the equation -1 + 4*d - 4*d^2 + 8*d^3 = 0. - Vaclav Kotesovec, May 01 2014