A123199 - OEIS

%I #30 Jul 16 2021 02:19:33

%S 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,

%T 40,-30,-68,30,40,-35,10,-1,1,12,54,100,15,-168,-76,168,15,-100,54,

%U -12,1,1,14,77,196,161,-238,-427,184,427,-238,-161,196,-77,14

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

%C 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).

%D Gengzhe Chang and Thomas W. Sederberg, Over and Over Again, The Mathematical Association of America, 1997, p. 164, figure 26.1.

%D Henry McKean and Victor Moll, Elliptic Curves: Function Theory, Geometry, Arithmetic, Cambridge University Press, 1997, p. 106, figure 2.22.

%H G. C. Greubel, <a href="/A123199/b123199.txt">Rows n = 0..40 of the irregular triangle, flattened</a>

%F Row n is made of coefficients of: (1 + 2*x - x^2)^n. - _Thomas Baruchel_, Jan 15 2015

%F From _Franck Maminirina Ramaharo_, Oct 13 2018: (Start)

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

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

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

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

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

%e Triangle begins:

%e     1;

%e     1,  2, -1;

%e     1,  4,  2, -4,   1;

%e     1,  6,  9, -4,  -9,   6, -1;

%e     1,  8, 20,  8, -26,  -8, 20, -8,   1;

%e     1, 10, 35, 40, -30, -68, 30, 40, -35, 10, -1;

%e     ...

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

%o (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 */

%o (Sage)

%o def T(n): return ( (1+2*x-x^2)^n ).full_simplify().coefficients(sparse=False)

%o [T(n) for n in (0..12)] # _G. C. Greubel_, Jul 15 2021

%Y Row sums: A000079 (powers of 2).

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

%K sign,tabf

%O 0,3

%A _Roger L. Bagula_, Oct 04 2006

%E New name from _Thomas Baruchel_, Jan 15 2015

%E Edited, and offset corrected by _Franck Maminirina Ramaharo_, Oct 13 2018