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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.

%I A105475 #24 Oct 08 2022 06:33:33
%S A105475 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,
%T A105475 16,60,122,130,66,14,1,1,16,82,208,295,232,91,16,1,2,21,108,324,582,
%U A105475 621,378,120,18,1,1,20,135,480,1035,1404,1176,576,153,20,1,2,26,170,675
%N 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.
%C A105475 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
%C A105475 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
%C A105475 Also the convolution triangle of A000034. - _Peter Luschny_, Oct 08 2022
%H A105475 Alois P. Heinz, <a href="/A105475/b105475.txt">Rows n = 1..141, flattened</a>
%e A105475 T(4,2) = 6 because we have (1,3), (3,1), (2,2), (2,2'), (2',2) and (2',2').
%e A105475 Triangle begins:
%e A105475 1;
%e A105475 2, 1;
%e A105475 1, 4,  1;
%e A105475 2, 6,  6, 1;
%e A105475 1, 8, 15, 8, 1;
%e A105475 Triangle (0, 2, -3/2, -1/2, 0, 0, 0...) DELTA (1, 0, 0, 0, 0, ...) begins:
%e A105475 1
%e A105475 0, 1
%e A105475 0, 2, 1
%e A105475 0, 1, 4, 1
%e A105475 0, 2, 6, 6, 1
%e A105475 0, 1, 8, 15, 8, 1
%e A105475 0, 2, 11, 26, 28, 10, 1
%e A105475 0, 1, 12, 42, 64, 45, 12, 1
%p A105475 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
%p A105475 # second Maple program:
%p A105475 b:= proc(n) option remember; `if`(n=0, 1,
%p A105475       expand(add((2-irem(i, 2))*b(n-i)*x, i=1..n)))
%p A105475     end:
%p A105475 T:= n-> (p-> seq(coeff(p, x, k), k=1..n))(b(n)):
%p A105475 seq(T(n), n=1..14);  # _Alois P. Heinz_, Oct 16 2013
%p A105475 # Uses function PMatrix from A357368. Adds a row above and a column to the left.
%p A105475 PMatrix(10, n -> [1, 2][irem(n-1, 2) + 1]); # _Peter Luschny_, Oct 08 2022
%t A105475 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 *)
%Y A105475 Row sums yield A105476.
%Y A105475 Cf. Diagonals: A000007, A000034, A000012, A005843, A000384, A100504.
%K A105475 nonn,tabl
%O A105475 1,2
%A A105475 _Emeric Deutsch_, Apr 09 2005