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A305098 Triangle read by rows: T(0,0) = 1; T(n,k) = -T(n-1,k) + 2 T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0. 4
1, -1, 1, 2, -1, -4, 1, 6, 4, -1, -8, -12, 1, 10, 24, 8, -1, -12, -40, -32, 1, 14, 60, 80, 16, -1, -16, -84, -160, -80, 1, 18, 112, 280, 240, 32, -1, -20, -144, -448, -560, -192, 1, 22, 180, 672, 1120, 672, 64, -1, -24, -220, -960, -2016, -1792, -448 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A303872 ((-1+2*x)^n).
The coefficients in the expansion of 1/(1+x-2x^2) are given by the sequence generated by the row sums.
When n is even the numbers in the row are positive, and when n is odd the numbers in the row are negative.
REFERENCES
Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 389-391.
LINKS
FORMULA
G.f.: 1 / (1 + t*x - 2t^2).
EXAMPLE
Triangle begins:
1;
-1;
1, 2;
-1, -4;
1, 6, 4;
-1, -8, -12;
1, 10, 24, 8;
-1, -12, -40, -32;
1, 14, 60, 80, 16;
-1, -16, -84, -160, -80;
1, 18, 112, 280, 240, 32;
-1, -20, -144, -448, -560, -192;
1, 22, 180, 672, 1120, 672, 64;
-1, -24, -220, -960, -2016, -1792, -448;
1, 26, 264, 1320, 3360, 4032, 1792, 128;
-1, -28, -312, -1760, -5280, -8064, -5376, -1024;
1, 30, 364, 2288, 7920, 14784, 13440, 4608, 256;
-1, -32, -420, -2912, -11440, -25344, -29568, -15360, -2304;
MATHEMATICA
t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, -t[n - 1, k] + 2 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten
PROG
(PARI) T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, -T(n-1, k) + 2*T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 26 2018
CROSSREFS
Signed version of A128099.
Row sums give A077925.
Cf. A303872, A033999 (column 0).
Sequence in context: A146938 A147418 A146386 * A128099 A182242 A261605
KEYWORD
tabf,easy,sign
AUTHOR
Shara Lalo, May 25 2018
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)