OFFSET
0,8
COMMENTS
Row sums are one.
LINKS
G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened
FORMULA
T(n, k) = coefficients of the expansion of p(x, n), where p(x,n) = (1+x-x^3)^(n + 1)*Sum_{j >= 0} (j+1)^n*(-x + x^3)^j.
T(n, 1) = (-1)*A000295(n) for n >= 2.
EXAMPLE
Irregular triangle begins as:
1;
1;
1, -1, 0, 1;
1, -4, 1, 4, -2, 0, 1;
1, -11, 11, 10, -22, 3, 11, -3, 0, 1;
1, -26, 66, 0, -131, 78, 62, -78, 6, 26, -4, 0, 1;
1, -57, 302, -245, -547, 905, 74, -901, 342, 292, -228, 10, 57, -5, 0, 1;
MATHEMATICA
p[x_, n_] = (1+x-x^3)^(n+1)*Sum[(j+1)^n*(-x+x^3)^j, {j, 0, Infinity}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten
PROG
(Sage)
def T(n, k): return ( (1+x-x^3)^(n+1)*sum((j+1)^n*(x^3-x)^j for j in (0..3*n+1)) ).series(x, 3*n+3).list()[k]
flatten([1]+[[T(n, k) for k in (0..3*n-3)] for n in (1..12)]) # G. C. Greubel, Jan 06 2022
CROSSREFS
KEYWORD
tabf,sign
AUTHOR
Roger L. Bagula, Feb 17 2009
EXTENSIONS
Edited by G. C. Greubel, Jan 06 2022
STATUS
approved