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A304213 Triangle read by rows: T(0,0) = 1; T(n,k) = 2*T(n-1,k) - 2*T(n-2,k-1) + T(n-3,k-2) for k = 0..floor(2*n/3); T(n,k)=0 for n or k < 0.

%I A304213 #25 Sep 05 2018 02:29:07
%S A304213 1,2,4,-2,8,-8,1,16,-24,8,32,-64,36,-4,64,-160,128,-32,1,128,-384,400,
%T A304213 -160,18,256,-896,1152,-640,136,-6,512,-2048,3136,-2240,720,-80,1,
%U A304213 1024,-4608,8192,-7168,3120,-592,32,2048,-10240,20736,-21504,11872,-3264,360,-8,4096,-22528,51200,-61440,41216,-15008,2624,-160,1
%N A304213 Triangle read by rows: T(0,0) = 1; T(n,k) = 2*T(n-1,k) - 2*T(n-2,k-1) + T(n-3,k-2) for k = 0..floor(2*n/3); T(n,k)=0 for n or k < 0.
%C A304213 The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A304209.
%C A304213 The coefficients in the expansion of 1/(1-2*x+2*x^2-x^3) are given by the sequence generated by the row sums.
%D A304213 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 204, 205.
%H A304213 Shara Lalo, <a href="/A304213/a304213.pdf">Left-justified triangle</a>
%e A304213 Triangle begins:
%e A304213       1;
%e A304213       2;
%e A304213       4,      -2;
%e A304213       8,      -8,      1;
%e A304213      16,     -24,      8;
%e A304213      32,     -64,     36,      -4;
%e A304213      64,    -160,    128,     -32,      1;
%e A304213     128,    -384,    400,    -160,     18;
%e A304213     256,    -896,   1152,    -640,    136,      -6;
%e A304213     512,   -2048,   3136,   -2240,    720,     -80,     1;
%e A304213    1024,   -4608,   8192,   -7168,   3120,    -592,    32;
%e A304213    2048,  -10240,  20736,  -21504,  11872,   -3264,   360,     -8;
%e A304213    4096,  -22528,  51200,  -61440,  41216,  -15008,  2624,   -160,   1;
%e A304213    8192,  -49152, 123904, -168960, 133632,  -60928, 14896,  -1632,  50;
%e A304213   16384, -106496, 294912, -450560, 410880, -225792, 71680, -11776, 780, -10;
%e A304213   ...
%o A304213 (PARI) T(n,k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 2*T(n-1,k)-2*T(n-2,k-1)+T(n-3,k-2)));
%o A304213 tabf(nn) = for (n=0, nn, for (k=0, 2*n\3, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, May 10 2018
%Y A304213 Row sums is similar to A021823.
%Y A304213 Cf. A304209.
%K A304213 tabf,easy,sign
%O A304213 0,2
%A A304213 _Shara Lalo_, May 08 2018