A172099 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = x^(2*n-1)*p(n-1, x) + p(n-2, x) with p(0, x) = 1 and p(1, x) = 1 + x, read by rows.
1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1
Offset: 0
Examples
Irregular triangle begins as: 1; 1, 1; 1, 0, 0, 1, 1; 1, 1, 0, 0, 0, 1, 0, 0, 1, 1; 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1; 1, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1;
Links
- G. C. Greubel, Rows n = 0..30 of the irregular triangle, flattened
Programs
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Mathematica
p[n_, x_]:= p[n, x]= If[n<2, n*x+1, x^(2*n-1)*p[n-1, x] + p[n-2, x]]; Table[CoefficientList[p[n, x], x], {n,0,10}]//Flatten -
SageMath
@CachedFunction def p(n,x): if (n<2): return n*x+1 else: return x^(2*n-1)*p(n-1, x) + p(n-2, x) def T(n,k): return ( p(n,x) ).series(x, n^2+2).list()[k] flatten([[T(n,k) for k in (0..n^2)] for n in (0..12)]) # G. C. Greubel, Apr 07 2022
Formula
T(n, k) = [x^k]( p(n, x) ), where p(n, x) = x^(2*n-1)*p(n-1, x) + p(n-2, x) with p(0, x) = 1 and p(1, x) = 1 + x.
Sum_{k=0..n^2} T(n, k) = Fibonacci(n+2) (row sums). - G. C. Greubel, Apr 07 2022
Extensions
Edited by G. C. Greubel, Apr 07 2022
Comments