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A172098
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = x^(n-1)*p(n-1, x) + p(n-2, x) with p(0, x) = 1 and p(1, x) = 1 + x, read by rows.
2
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1
OFFSET
0,24
COMMENTS
Rows have A089071(n) number of terms, i.e., 0 <= k <= A089071(n) - 1, for n >= 0. - G. C. Greubel, Apr 07 2022
FORMULA
T(n, k) = [x^k]( p(n, x) ), where p(n, x) = x^(n-1)*p(n-1, x) + p(n-2, x) with p(0, x) = 1 and p(1, x) = 1 + x.
Sum_{k=0..A089071(n)-1} T(n, k) = Fibonacci(n+2) (row sums). - G. C. Greubel, Apr 07 2022
EXAMPLE
Irregular triangle begins as:
1;
1, 1;
1, 1, 1;
1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1;
MATHEMATICA
p[n_, x_]:= p[n, x]= If[n<2, n*x+1, x^(n-1)*p[n-1, x] + p[n-2, x]];
Table[CoefficientList[p[n, x], x], {n, 0, 12}]//Flatten
PROG
(SageMath)
@CachedFunction
def p(n, x):
if (n<2): return n*x+1
else: return x^(n-1)*p(n-1, x) + p(n-2, x)
def T(n, k): return ( p(n, x) ).series(x, (n^2-n+4-2*0^n)/2+1).list()[k]
flatten([[T(n, k) for k in (0..(n^2-n+2-2*0^n)/2)] for n in (0..12)]) # G. C. Greubel, Apr 07 2022
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Roger L. Bagula, Jan 25 2010
EXTENSIONS
Edited by G. C. Greubel, Apr 07 2022
STATUS
approved