

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
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OFFSET

1,2


COMMENTS

Riordan array ((1+2x)/(1x^2),x(1+2x)/(1x^2)). Factorizes as ((1+2x)/(1x^2),x)*(1,x(1+2x)/(1x^2)). Row sums A105476 form an eigensequence for ((1+2x)/(1x^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)/(1t*zz^22*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((2irem(i, 2))*b(ni)*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 (* JeanFrançois 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



