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
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
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