login
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, 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.
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
KEYWORD
nonn,tabf
AUTHOR
Roger L. Bagula, Dec 10 2009
EXTENSIONS
Edited by G. C. Greubel, Jul 18 2021
STATUS
approved