A003480 -id:A003480 - cp's OEIS frontend

A113859 Expansion of (7-14x+6x^2)/((1-x)(2x^2-4*x+1)); related to the binomial transform of Pell numbers A000129 (see formula and comment for A007070).

7, 21, 69, 233, 793, 2705, 9233, 31521, 107617, 367425, 1254465, 4283009, 14623105, 49926401, 170459393, 581984769, 1987020289, 6784111617, 23162405889, 79081400321, 270000789505, 921840357377, 3147359850497, 10745758687233

Offset: 0

Creighton Dement, Jan 25 2006

If g.f. (x^6+5*x^4+6*x^2+1)/(x^7+6*x^5+10*x^3+4*x) is expanded, where (x^6+5*x^4+6*x^2+1) and (x^7+6*x^5+10*x^3+4*x) are the 7th and 8th Fibonacci polynomials, respectively, the sequence: [0, 7/8, 0, -21/16, 0, 69/32, 0, -233/64, 0, 793/128, 0, -2705/256, ] is returned. (a(n)) is seen to be the numerators of the bisection of this sequences, apart from signs.

Cf. A007070, A007052, A003480, A111567, A000129.

with(combinat, fibonacci): seq(fibonacci(i, x), i=1..15); [[generates sequence of Fibonacci polynomials]]

a(n+1) - a(n) = A007070(n+2), a(n) - 2*a(n+1) + a(n+2) = A007052(n+3) (Number of order consecutive partitions of n), a(n+3) - 3*a(n+2) + 3*a(n+1) - a(n) = A003480(n+4), a(n+2) - a(n) = A111567(n+3)

A181307 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with only nonzero entries (0<=k<=floor(n/2)).

Original entry on oeis.org

1, 2, 6, 1, 18, 6, 54, 27, 1, 162, 108, 10, 486, 405, 64, 1, 1458, 1458, 334, 14, 4374, 5103, 1549, 117, 1, 13122, 17496, 6652, 760, 18, 39366, 59049, 27064, 4238, 186, 1, 118098, 196830, 105796, 21324, 1450, 22, 354294, 649539, 401041, 99646, 9480, 271

Offset: 0

Emeric Deutsch, Oct 13 2010

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.

Number of entries in row n is 1+floor(n/2).

			T(2,1) = 1 because we have (1/1) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
  1;
  2;
  6,1;
  18,6;
  54,27,1;
  162,108,10;

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.

Cf. A003480, A008776, A054146.

G := (1-z)^2/(1-4*z+3*z^2-t*z^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

G.f.: G(t,z) = (1-z)^2/(1-4*z+3*z^2-t*z^2).
G.f. of column k: z^(2*k)/((1-3*z)^(k+1)*(1-z)^(k-1)) (we have a Riordan array).
Sum_{k>=0} T(n,k) = A003480(n).
T(n,0) = 2*3^(n-1) = A008776(n-1).
Sum_{k>=0} k*T(n,k) = A054146(n-1).

cp's OEIS Frontend

A113859 Expansion of (7-14x+6x^2)/((1-x)(2x^2-4*x+1)); related to the binomial transform of Pell numbers A000129 (see formula and comment for A007070).

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A181307 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with only nonzero entries (0<=k<=floor(n/2)).

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cp's OEIS Frontend

A113859 Expansion of (7-14*x+6*x^2)/((1-x)*(2*x^2-4*x+1)); related to the binomial transform of Pell numbers A000129 (see formula and comment for A007070).

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Maple

Formula

A181307 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with only nonzero entries (0<=k<=floor(n/2)).

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Author

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Maple

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A113859 Expansion of (7-14x+6x^2)/((1-x)(2x^2-4*x+1)); related to the binomial transform of Pell numbers A000129 (see formula and comment for A007070).