A181366 Least entry in a 2-composition of n, summed over all 2-compositions of n. A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.
0, 1, 2, 5, 10, 20, 38, 74, 144, 282, 552, 1086, 2144, 4247, 8430, 16761, 33364, 66479, 132566, 264520, 528078, 1054636, 2106854, 4209853, 8413548, 16817253, 33618758, 67212301, 134384182, 268703498, 537302782, 1074437977, 2148606246
Offset: 1
Keywords
Examples
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.
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.
Crossrefs
Cf. A181365.
Programs
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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);
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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}]; Array[a, terms] (* Jean-François Alcover, Nov 11 2020, after Alois P. Heinz in A181365 *)
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
Comments