OFFSET
0,5
COMMENTS
Row sums of triangle in A189073.
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 0..1000
M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
S. Heubach, A. Knopfmacher, M. E. Mays and A. Munagi, Inversions in Compositions of Integers, Quaest. Math. 34 (2011), no. 2, 187-202.
Index entries for linear recurrences with constant coefficients, signature (5,-6,-4,8)
FORMULA
a(n) = 2^(n-1)*(1/24*(n+2)*(n+1)-5/36*(n+1)-5/108)-2/27*(-1)^n for n>0.
a(n) = +5*a(n-1) -6*a(n-2) -4*a(n-3) +8*a(n-4).
G.f.: x^3*(1-x)/((1+x)*(1-2*x)^3).
EXAMPLE
a(4)=4. There are eight compositions of 4. Five of these (the partitions of 4) have no inversions. The remaining three: 3+1, 2+1+1, 1+2+1 have 1,2,1 inversions respectively. - Geoffrey Critzer, Mar 19 2014
MAPLE
with(PolynomialTools):n:=33:taypoly:=taylor(x^3*(1-x)/((1+x)*(1-2*x)^3), x=0, n+1):seq(coeff(taypoly, x, m), m=0..n); # Nathaniel Johnston, Apr 17 2011
# second Maple program:
a:= n-> `if`(n=0, 0, (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>,
<8|-4|-6|5>>^n. <<-1/8, 0, 0, 1>>)[1, 1]):
seq(a(n), n=0..40); # Alois P. Heinz, Apr 04 2016
MATHEMATICA
nn=30; CoefficientList[Series[(1-x)*x^3/((1+x)*(1-x-x)^3), {x, 0, nn}], x] (* Geoffrey Critzer, Mar 19 2014 *)
LinearRecurrence[{5, -6, -4, 8}, {0, 0, 0, 1, 4}, 40] (* Harvey P. Dale, May 25 2016 *)
PROG
(PARI) A189052(n)=2^(n-1)*(1/24*(n+2)*(n+1)-5/36*(n+1)-5/108)-2/27*(-1)^n;
vector(33, n, A189052(n)) /* show terms */ /* Joerg Arndt, Apr 16 2011 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Apr 16 2011
STATUS
approved