OFFSET
1,3
COMMENTS
a(n) = Sum(k*A181365(n,k), k>=0).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..500
G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741.
FORMULA
a(n) = Sum_{k>=0} k*A181365(n,k).
G.f. for 2-compositions with all entries >= k is h(k,z)=(1-z)^2/(1-2z+z^2-z^{2k}) if k>0 and h(0,z)=(1-z)^2/(1-4z+2z^2) if k=0.
G.f. for 2-compositions with least entry k is f(k,z)=h(k,z)-h(k+1,z).
G.f.: G(z) = Sum(k*f(k,z), k=1..infinity).
a(n) ~ 2^(n-2). - Vaclav Kotesovec, Sep 03 2014
EXAMPLE
a(2)=1 because the 2-compositions of 2, written as (top row / bottom row), are (1/1), (0/2), (2/0), (1,0/0,1), (0,1/1,0), (1,1/0,0), (0,0/1,1) and the least entries are 1 and eight 0's.
MAPLE
h := proc (k) if k = 0 then (1-z)^2/(1-4*z+2*z^2) else (1-z)^2/(1-2*z+z^2-z^(2*k)) end if end proc: f := proc (k) options operator, arrow; h(k)-h(k+1) end proc: G := sum(k*f(k), k = 1 .. 50): Gser := series(G, z = 0, 45): seq(coeff(Gser, z, n), n = 1 .. 35);
MATHEMATICA
terms = 100;
A[n_, k_] := A[n, k] = If[n==0, 1, Sum[If[i==0 && j==0, 0, A[n-i-j, k]], {j, k, n}, {i, k, n-j}]];
T[n_, k_] := A[n, k] - A[n, k+1];
a[n_] := Sum[k T[n, k], {k, 0, terms}];
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 15 2010
STATUS
approved