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A123199
Irregular triangle read by rows: row n is the expansion of (1 + 2*x - x^2)^n.
13
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.
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).
Sequence in context: A122517 A256098 A177276 * A212282 A346032 A157125
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