A171502 Irregular triangle T(n, k) = coefficients of p(x, n), where p(x, n) = ((1-x^(2*n))/(1-x))*p(x, n-1), p(x, 0) = 1, and p(x, 1) = (1-x^4)/(1-x), read by rows.

1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 3, 2, 1, 1, 3, 6, 10, 14, 18, 21, 23, 23, 21, 18, 14, 10, 6, 3, 1, 1, 4, 10, 20, 34, 52, 73, 96, 119, 140, 157, 168, 172, 168, 157, 140, 119, 96, 73, 52, 34, 20, 10, 4, 1

Offset

1,7

References

Samuel I. Goldberg, Curvature and Homology, Dover, New York, 1998, page 144.

Links

G. C. Greubel, Rows n = 1..20 of the irregular triangle, flattened

FindStat - Combinatorial Statistic Finder, The inversion number of a standard Young tableau as defined by Haglund and Stevens.

Formula

T(n, k) = coefficients of p(x, n), where p(x, n) = ((1-x^(2*n))/(1-x))*p(x, n-1), p(x, 0) = 1, and p(x, 1) = (1-x^4)/(1-x).

T(n, k) = coefficients of f(n, x), where f(n, x) = (1/(1-x)^(n-1))*Product_{j=2..n} (1 - x^(2*j)). - G. C. Greubel, Jul 18 2021

Example

Irregular triangle begins as:
  1;
  1, 1, 1,  1;
  1, 2, 3,  4,  4,  4,  3,  2,  1;
  1, 3, 6, 10, 14, 18, 21, 23, 23, 21, 18, 14, 10, 6, 3, 1;

Mathematica

p[n_, x_]:= p[n, x]= Product[1 - x^(2*j), {j, 2, n}]/(1-x)^(n-1);
T[n_]:= T[n] = CoefficientList[p[n, x], x];
Table[T[n], {n, 10}]//Flatten

Prog

(Sage)
@CachedFunction
def p(n, x): return product( 1-x^(2*j) for j in (2..n) )/(1-x)^(n-1)
def T(n): return ( p(n, x) ).full_simplify().coefficients(sparse=False)
flatten([T(n) for n in (1..12)]) # G. C. Greubel, Jul 18 2021

Crossrefs

Cf. A142724.

Sequence in context: A263408 A109870 A364368 * A005102 A030241 A062750

Adjacent sequences: A171499 A171500 A171501 * A171503 A171504 A171505

Keyword

nonn,tabf

Author

Roger L. Bagula, Dec 10 2009

Extensions

Edited by G. C. Greubel, Jul 18 2021

Status

approved