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A105475 Triangle read by rows: T(n,k) is number of compositions of n into k parts when each even part can be of two kinds. 3
1, 2, 1, 1, 4, 1, 2, 6, 6, 1, 1, 8, 15, 8, 1, 2, 11, 26, 28, 10, 1, 1, 12, 42, 64, 45, 12, 1, 2, 16, 60, 122, 130, 66, 14, 1, 1, 16, 82, 208, 295, 232, 91, 16, 1, 2, 21, 108, 324, 582, 621, 378, 120, 18, 1, 1, 20, 135, 480, 1035, 1404, 1176, 576, 153, 20, 1, 2, 26, 170, 675 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Riordan array ((1+2x)/(1-x^2),x(1+2x)/(1-x^2)). Factorizes as ((1+2x)/(1-x^2),x)*(1,x(1+2x)/(1-x^2)). Row sums A105476 form an eigensequence for ((1+2x)/(1-x^2),x). - Paul Barry, Feb 10 2011

Triangle T(n,k), 1<=k<=n, given by (0, 2, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 18 2012

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

EXAMPLE

T(4,2) = 6 because we have (1,3), (3,1), (2,2), (2,2'), (2',2) and (2',2').

Triangle begins:

1;

2, 1;

1, 4,  1;

2, 6,  6, 1;

1, 8, 15, 8, 1;

Triangle (0, 2, -3/2, -1/2, 0, 0, 0...) DELTA (1, 0, 0, 0, 0, ...) begins:

1

0, 1

0, 2, 1

0, 1, 4, 1

0, 2, 6, 6, 1

0, 1, 8, 15, 8, 1

0, 2, 11, 26, 28, 10, 1

0, 1, 12, 42, 64, 45, 12, 1

MAPLE

G:=t*z*(1+2*z)/(1-t*z-z^2-2*t*z^2): Gser:=simplify(series(G, z=0, 14)): for n from 1 to 12 do P[n]:=sort(coeff(Gser, z^n)) od: for n from 1 to 12 do seq(coeff(P[n], t^k), k=1..n) od; # yields sequence in triangular form

# second Maple program:

b:= proc(n) option remember; `if`(n=0, 1,

      expand(add((2-irem(i, 2))*b(n-i)*x, i=1..n)))

    end:

T:= n-> (p-> seq(coeff(p, x, k), k=1..n))(b(n)):

seq(T(n), n=1..14);  # Alois P. Heinz, Oct 16 2013

MATHEMATICA

max = 14; g = t*z*(1 + 2*z)/(1 - t*z - z^2 - 2*t*z^2); gser = Series[g, {z, 0, max}]; coes = CoefficientList[gser, {z, t}]; Table[ Table[ coes[[n, k]], {k, 2, n}], {n, 2, max}] // Flatten (* Jean-François Alcover, Oct 02 2013, after Maple *)

CROSSREFS

Row sums yield A105476.

Cf. Diagonals: A000007, A000034, A000012, A005843, A000384, A100504.

Sequence in context: A193554 A131350 A131087 * A249061 A210209 A328649

Adjacent sequences:  A105472 A105473 A105474 * A105476 A105477 A105478

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Apr 09 2005

STATUS

approved

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Last modified December 6 19:22 EST 2019. Contains 329809 sequences. (Running on oeis4.)