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

%I A123199 #30 Jul 16 2021 02:19:33
%S A123199 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 A123199 40,-30,-68,30,40,-35,10,-1,1,12,54,100,15,-168,-76,168,15,-100,54,
%U A123199 -12,1,1,14,77,196,161,-238,-427,184,427,-238,-161,196,-77,14
%N A123199 Irregular triangle read by rows: row n is the expansion of (1 + 2*x - x^2)^n.
%C A123199 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 A123199 Gengzhe Chang and Thomas W. Sederberg, Over and Over Again, The Mathematical Association of America, 1997, p. 164, figure 26.1.
%D A123199 Henry McKean and Victor Moll, Elliptic Curves: Function Theory, Geometry, Arithmetic, Cambridge University Press, 1997, p. 106, figure 2.22.
%H A123199 G. C. Greubel, <a href="/A123199/b123199.txt">Rows n = 0..40 of the irregular triangle, flattened</a>
%F A123199 Row n is made of coefficients of: (1 + 2*x - x^2)^n. - _Thomas Baruchel_, Jan 15 2015
%F A123199 From _Franck Maminirina Ramaharo_, Oct 13 2018: (Start)
%F A123199 G.f.: 1/(1 - (1 +  2*x - x^2)*y).
%F A123199 E.g.f.: exp((1 +  2*x - x^2)*y).
%F A123199 T(n,1) = A005843(n).
%F A123199 T(n,2) = A014107(n).
%F A123199 T(n,n) = A098335(n). (End)
%e A123199 Triangle begins:
%e A123199     1;
%e A123199     1,  2, -1;
%e A123199     1,  4,  2, -4,   1;
%e A123199     1,  6,  9, -4,  -9,   6, -1;
%e A123199     1,  8, 20,  8, -26,  -8, 20, -8,   1;
%e A123199     1, 10, 35, 40, -30, -68, 30, 40, -35, 10, -1;
%e A123199     ...
%t A123199 Table[CoefficientList[(-x^2 + 2*x + 1)^n, x], {n, 0, 10}]//Flatten
%o A123199 (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 A123199 (Sage)
%o A123199 def T(n): return ( (1+2*x-x^2)^n ).full_simplify().coefficients(sparse=False)
%o A123199 [T(n) for n in (0..12)] # _G. C. Greubel_, Jul 15 2021
%Y A123199 Row sums: A000079 (powers of 2).
%Y A123199 Cf. A122753, A123018, A123019, A123021, A123027, A123202, A123217, A123221.
%K A123199 sign,tabf
%O A123199 0,3
%A A123199 _Roger L. Bagula_, Oct 04 2006
%E A123199 New name from _Thomas Baruchel_, Jan 15 2015
%E A123199 Edited, and offset corrected by _Franck Maminirina Ramaharo_, Oct 13 2018