A303872 Triangle read by rows: T(0,0) = 1; T(n,k) = -T(n-1,k) + 2 T(n-1,k-1) for k = 0,1,...,n; T(n,k)=0 for n or k < 0.

1, -1, 2, 1, -4, 4, -1, 6, -12, 8, 1, -8, 24, -32, 16, -1, 10, -40, 80, -80, 32, 1, -12, 60, -160, 240, -192, 64, -1, 14, -84, 280, -560, 672, -448, 128, 1, -16, 112, -448, 1120, -1792, 1792, -1024, 256, -1, 18, -144, 672, -2016, 4032, -5376, 4608, -2304, 512

Offset: 0

Author

Shara Lalo, May 25 2018

Keywords

• tabl
• easy
• sign

Comments

Row n gives coefficients in expansion of (-1+2x)^n. Row sums=1.

In the center-justified triangle, the numbers in skew diagonals pointing top-Left give the triangle in A133156 (coefficients of Chebyshev polynomials of the second kind), and the numbers in skew diagonals pointing top-right give the triangle in A305098. The coefficients in the expansion of 1/(1-x) are given by the sequence generated by the row sums. The generating function of the central terms is 1/sqrt(1+8x), signed version of A059304.

Examples

			Triangle begins:
   1;
  -1,   2;
   1,  -4,   4;
  -1,   6, -12,    8;
   1,  -8,  24,  -32,   16;
  -1,  10, -40,   80,  -80,    32;
   1, -12,  60, -160,  240,  -192,   64;
  -1,  14, -84,  280, -560,   672, -448,   128;
   1, -16, 112, -448, 1120, -1792, 1792, -1024, 256;
		

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

• Shara Lalo, Skew diagonals in center-justified triangle
• Paweł Lorek and Piotr Markowski, Absorption time and absorption probabilities for a family of multidimensional gambler models, arXiv:1812.00690 [math.PR], 2018.

Crossrefs

Row sums give A000012.

Signed version of A013609 ((1+2*x)^n).

Cf. A033999 (column 0).

Programs

Mathematica

T[0, 0] = 1; T[n_, k_] := If[n < 0 || k < 0, 0, - T[n - 1, k] + 2 T[n - 1, k - 1]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
For[i = 0, i < 4, i++, Print[CoefficientList[Expand[(-1 +2 x)^i], x]]]

PARI

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, -T(n-1, k) + 2*T(n-1, k-1)));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 26 2018

Formula

G.f.: 1 / (1 + t - 2t*x).

T(n,k) = (-1)^(n+k)*2^k*binomial(n,k). - Stefano Spezia, Aug 08 2025