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%I #19 Jul 15 2021 04:37:49
%S 1,1,1,1,-1,1,2,3,-5,1,3,20,-32,9,1,4,58,-82,5,15,1,5,125,-108,-161,
%T 170,-31,1,6,229,17,-797,603,7,-65,1,7,378,532,-2210,664,1468,-968,
%U 129,1,8,580,1820,-4226,-2846,8788,-4388,9,255,1,9,843,4440,-5262
%N Triangle read by rows: the n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A123162(n,j)*x^j*(1 - x)^(n - j).
%H G. C. Greubel, <a href="/A123217/b123217.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%F From _Franck Maminirina Ramaharo_, Oct 10 2018: (Start)
%F Row n = coefficients in the expansion of (1-x)^n + x*((1 - 2*sqrt((1-x)*x))^n*(1 - x + sqrt((1-x)*x)) - (1-x - sqrt((1-x)*x))*(1 + 2*sqrt((1-x)*x))^n)/(2*sqrt((1 - x)*x)*(2*x-1)).
%F G.f.: (1 - (2 - x)*y + (1 - 4*x + 3*x^2)*y^2 - (x - 3*x^2 + 2*x^3)*y^3)/(1 - (3 - x)*y + (3 - 6*x + 4*x^2)*y^2 - (1 - 5*x + 8*x^2 - 4*x^3)*y^3).
%F E.g.f.: exp((1 - x)*y) + x*((1 - x + sqrt((1 - x)*x))*exp((1 - 2*sqrt((1 - x)*x))*y) - (1 - x - sqrt((1 - x)*x))*exp((1 + 2*sqrt((1 - x)*x))*y))/(2*(2*x - 1)*sqrt((1 - x)*x)) - (1 - 3*x)/(1 - 2*x) + 1. (End)
%e Triangle begins:
%e 1;
%e 1;
%e 1, 1, -1;
%e 1, 2, 3, -5;
%e 1, 3, 20, -32, 9;
%e 1, 4, 58, -82, 5, 15;
%e 1, 6, 229, 17, -797, 603, 7, -65;
%e 1, 7, 378, 532, -2210, 664, 1468, -968, 129;
%e 1, 8, 580, 1820, -4226, -2846, 8788, -4388, 9, 255;
%e ... reformatted and extended. - _Franck Maminirina Ramaharo_, Oct 10 2018
%t t[n_, k_]= If[k==0, 1, Binomial[2*n-1, 2*k-1]];
%t p[n_,x_]:= p[n,x]= Sum[t[n,j]*x^j*(1-x)^(n-j), {j,0,n}];
%t Table[CoefficientList[p[n,x], x], {n, 0, 10}]//Flatten
%o (Maxima) A123162(n, k) := if n = 0 and k = 0 or k = 0 then 1 else binomial(2*n - 1, 2*k - 1)$
%o P(x, n) := expand(sum(A123162(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$
%o T(n, k) := ratcoef(P(x, n), x, k)$
%o tabf(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, hipow(P(x, n), x)))$ /* _Franck Maminirina Ramaharo_, Oct 10 2018 */
%o (Sage)
%o def b(n,k): return 1 if (k==0) else binomial(2*n-1, 2*k-1)
%o def p(n,x): return sum( b(n,j)*x^j*(1-x)^(n-j) for j in (0..n) )
%o def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
%o [T(n) for n in (0..12)] # _G. C. Greubel_, Jul 15 2021
%Y Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123221.
%K tabf,sign
%O 0,7
%A _Roger L. Bagula_, Oct 04 2006
%E Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Oct 11 2018
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