A123199 Irregular triangle read by rows: row n is the expansion of (1 + 2*x - x^2)^n.

1, 1, 2, -1, 1, 4, 2, -4, 1, 1, 6, 9, -4, -9, 6, -1, 1, 8, 20, 8, -26, -8, 20, -8, 1, 1, 10, 35, 40, -30, -68, 30, 40, -35, 10, -1, 1, 12, 54, 100, 15, -168, -76, 168, 15, -100, 54, -12, 1, 1, 14, 77, 196, 161, -238, -427, 184, 427, -238, -161, 196, -77, 14

Offset: 0

Author

Roger L. Bagula, Oct 04 2006

Keywords

• sign
• tabf

Comments

The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A007318(n, j)*(2*x)^j*(1 - x^2)^(n-j).

Examples

			Triangle begins:
    1;
    1,  2, -1;
    1,  4,  2, -4,   1;
    1,  6,  9, -4,  -9,   6, -1;
    1,  8, 20,  8, -26,  -8, 20, -8,   1;
    1, 10, 35, 40, -30, -68, 30, 40, -35, 10, -1;
    ...
		

References

• Gengzhe Chang and Thomas W. Sederberg, Over and Over Again, The Mathematical Association of America, 1997, p. 164, figure 26.1.
• Henry McKean and Victor Moll, Elliptic Curves: Function Theory, Geometry, Arithmetic, Cambridge University Press, 1997, p. 106, figure 2.22.

Links

• G. C. Greubel, Rows n = 0..40 of the irregular triangle, flattened

Crossrefs

Row sums: A000079 (powers of 2).

Cf. A122753, A123018, A123019, A123021, A123027, A123202, A123217, A123221.

Programs

Mathematica

Table[CoefficientList[(-x^2 + 2*x + 1)^n, x], {n, 0, 10}]//Flatten

Maxima

create_list(ratcoef((-x^2 + 2*x + 1)^n, x, k), n, 0, 10, k, 0, 2*n); /* Franck Maminirina Ramaharo, Oct 13 2018 */

Sage

def T(n): return ( (1+2*x-x^2)^n ).full_simplify().coefficients(sparse=False)
[T(n) for n in (0..12)] # G. C. Greubel, Jul 15 2021

Formula

Row n is made of coefficients of: (1 + 2*x - x^2)^n. - Thomas Baruchel, Jan 15 2015

From Franck Maminirina Ramaharo, Oct 13 2018: (Start)

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

E.g.f.: exp((1 + 2*x - x^2)*y).

T(n,1) = A005843(n).

T(n,2) = A014107(n).

T(n,n) = A098335(n). (End)

Extensions

New name from Thomas Baruchel, Jan 15 2015

Edited, and offset corrected by Franck Maminirina Ramaharo, Oct 13 2018