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A181338 Triangle read by rows: T(n,k) is the number of 2-compositions of n having largest entry k (1<=k<=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. 1
2, 5, 2, 12, 10, 2, 29, 41, 10, 2, 70, 152, 46, 10, 2, 169, 536, 193, 46, 10, 2, 408, 1830, 770, 198, 46, 10, 2, 985, 6120, 2972, 811, 198, 46, 10, 2, 2378, 20178, 11202, 3218, 816, 198, 46, 10, 2, 5741, 65867, 41481, 12484, 3259, 816, 198, 46, 10, 2, 13860 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The sum of entries in row n is A003480(n).

T(n,1)=A000129(n+1) (the Pell numbers).

Sum(k*T(n,k),k=0..n)=A181339.

REFERENCES

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.

LINKS

Table of n, a(n) for n=1..56.

FORMULA

G.f. for 2-compositions with all entries <= k is h(k,z)=(1-z)^2/(1-4z+2z^2+2z^{k+1}-z^{2k+2}).

G.f. for 2-compositions with largest entry k is f(k,z)=h(k,z)-h(k-1,z) (these are the column g.f.'s).

G.f. = G(t,z)=Sum(f(k,z)*t^k, k=1..infinity).

EXAMPLE

T(3,3)=2 because we have (0/3) and (3/0) (the 2-compositions are written as (top row/bottom row).

Triangle starts:

2;

5,2;

12,10,2;

29,41,10,2;

70,152,46,10,2;

MAPLE

h := proc (k) options operator, arrow: (1-z)^2/(1-4*z+2*z^2+2*z^(k+1)-z^(2*k+2)) end proc: f := proc (k) options operator, arrow; simplify(h(k)-h(k-1)) end proc: G := sum(f(k)*t^k, k = 1 .. 30): Gser := simplify(series(G, z = 0, 15)): for n to 11 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 11 do seq(coeff(P[n], t, k), k = 1 .. n) end do; # yields sequence in triangular form

CROSSREFS

Cf. A003480, A000129, A181339

Sequence in context: A240760 A207635 A205715 * A211175 A102469 A098886

Adjacent sequences:  A181335 A181336 A181337 * A181339 A181340 A181341

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Oct 15 2010

STATUS

approved

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Last modified October 13 23:25 EDT 2019. Contains 327983 sequences. (Running on oeis4.)