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.

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

Offset: 1

Author

Emeric Deutsch, Apr 09 2005

Keywords

• nonn
• tabl

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

Also the convolution triangle of A000034. - Peter Luschny, Oct 08 2022

Examples

			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
		

Links

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

Crossrefs

Row sums yield A105476.

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

Programs

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
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> [1, 2][irem(n-1, 2) + 1]); # Peter Luschny, Oct 08 2022

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 *)