A123217 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).

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, 170, -31, 1, 6, 229, 17, -797, 603, 7, -65, 1, 7, 378, 532, -2210, 664, 1468, -968, 129, 1, 8, 580, 1820, -4226, -2846, 8788, -4388, 9, 255, 1, 9, 843, 4440, -5262

Offset: 0

Author

Roger L. Bagula, Oct 04 2006

Keywords

• tabf
• sign

Examples

			Triangle begins:
  1;
  1;
  1, 1,  -1;
  1, 2,   3,   -5;
  1, 3,  20,  -32,     9;
  1, 4,  58,  -82,     5,    15;
  1, 6, 229,   17,  -797,   603,    7,   -65;
  1, 7, 378,  532, -2210,   664, 1468,  -968, 129;
  1, 8, 580, 1820, -4226, -2846, 8788, -4388,   9, 255;
  ... reformatted and extended. - _Franck Maminirina Ramaharo_, Oct 10 2018
		

Links

• G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened

Crossrefs

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123221.

Programs

Mathematica

t[n_, k_]= If[k==0, 1, Binomial[2*n-1, 2*k-1]];
p[n_,x_]:= p[n,x]= Sum[t[n,j]*x^j*(1-x)^(n-j), {j,0,n}];
Table[CoefficientList[p[n,x], x], {n, 0, 10}]//Flatten

Maxima

A123162(n, k) := if n = 0 and k = 0 or k = 0 then 1 else binomial(2*n - 1, 2*k - 1)$
P(x, n) := expand(sum(A123162(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$
T(n, k) := ratcoef(P(x, n), x, k)$
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 */

Sage

def b(n,k): return 1 if (k==0) else binomial(2*n-1, 2*k-1)
def p(n,x): return sum( b(n,j)*x^j*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
[T(n) for n in (0..12)] # G. C. Greubel, Jul 15 2021

Formula

From Franck Maminirina Ramaharo, Oct 10 2018: (Start)

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)).

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).

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)

Extensions

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 11 2018