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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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}];

Array[a, terms] (* Jean-Fran├žois Alcover, Nov 11 2020, after Alois P. Heinz in A181365 *)

CROSSREFS

Cf. A181365.

Sequence in context: A341581 A001629 A159230 * A068034 A222082 A327287

Adjacent sequences:  A181363 A181364 A181365 * A181367 A181368 A181369

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 15 2010

STATUS

approved

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Last modified April 21 17:36 EDT 2021. Contains 343156 sequences. (Running on oeis4.)