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

0,45

Comments

There are n^2 + 1 terms in row n, 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^(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

Example

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;

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

Prog

(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

Crossrefs

Cf. A000045, A002522, A172098.

Sequence in context: A332602 A306383 A376364 * A170957 A178725 A257402

Adjacent sequences: A172096 A172097 A172098 * A172100 A172101 A172102

Keyword

nonn,tabf

Author

Roger L. Bagula, Jan 25 2010

Extensions

Edited by G. C. Greubel, Apr 07 2022

Status

approved