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

0,3

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).

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

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)

Example

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;
    ...

Mathematica

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

Prog

(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

Crossrefs

Row sums: A000079 (powers of 2).

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

Sequence in context: A122517 A256098 A177276 * A212282 A346032 A157125

Adjacent sequences: A123196 A123197 A123198 * A123200 A123201 A123202

Keyword

sign,tabf

Author

Roger L. Bagula, Oct 04 2006

Extensions

New name from Thomas Baruchel, Jan 15 2015

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

Status

approved