A181299 -id:A181299 - cp's OEIS frontend

A181301 Number of 2-compositions of n having no column with equal entries. 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.

1, 2, 6, 20, 64, 206, 662, 2128, 6840, 21986, 70670, 227156, 730152, 2346942, 7543822, 24248256, 77941648, 250529378, 805281526, 2588432308, 8320049072, 26743297998, 85961510758, 276307781200, 888141556360, 2854770939522

Offset: 0

Emeric Deutsch, Oct 12 2010

a(n)=A181299(n,0).

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.

Cf. A181299.

g := (1+z)*(1-z)^2/(1-3*z-z^2+z^3): gser := series(g, z = 0, 30): seq(coeff(gser, z, n), n = 0 .. 27);

G.f. = (1+z)(1-z)^2/(1-3z-z^2+z^3).
a(n) = Sum_{k, 0<=k<=n} A060086(n,k)*2^k. - Philippe Deléham, Feb 24 2012
a(n) = 2*A033505(n-1), n>0. - R. J. Mathar, Jul 24 2022

A181300 Number of columns with top entry equal to bottom entry in all the 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.

Original entry on oeis.org

0, 0, 1, 4, 19, 80, 328, 1304, 5084, 19512, 73976, 277688, 1033848, 3822584, 14050824, 51385720, 187095240, 678570360, 2452626312, 8837584248, 31756892552, 113831195000, 407102551688, 1452956457336, 5175872174728

Offset: 0

Emeric Deutsch, Oct 12 2010

a(n)=Sum(k*A181299(n,k),k>=0).

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.

Index entries for linear recurrences with constant coefficients, signature (7,-12,-4,12,-4).

Cf. A181299

g := z^2*(1-z)^3/((1+z)*(1-4*z+2*z^2)^2): gser := series(g, z = 0, 30): seq(coeff(gser, z, n), n = 0 .. 27);

LinearRecurrence[{7,-12,-4,12,-4},{0,0,1,4,19,80},30] (* Harvey P. Dale, Apr 05 2023 *)

G.f. = z^2*(1-z)^3/[(1+z)(1-4z+2z^2)^2].

cp's OEIS Frontend

A181301 Number of 2-compositions of n having no column with equal entries. 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.

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