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.

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

Links

G. C. Greubel, Rows n = 0..30 of the irregular triangle, flattened

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

Cf. A000045, A089071, A172099.

Sequence in context: A249622 A287656 A043286 * A353801 A333254 A204162

Adjacent sequences: A172095 A172096 A172097 * A172099 A172100 A172101

Keyword

nonn,tabf

Author

Roger L. Bagula, Jan 25 2010

Extensions

Edited by G. C. Greubel, Apr 07 2022

Status

approved