OFFSET
0,7
COMMENTS
Row sums are equal to 1.
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^2)^(n + 1)*Sum_{j >= 0} (j+1)^n*(-x + x^2)^j.
T(n, 1) = (-1)*A000295(n) for n >= 2. - G. C. Greubel, Jan 06 2022
EXAMPLE
Irregular triangle begins as:
1;
1;
1, -1, 1;
1, -4, 5, -2, 1;
1, -11, 22, -23, 14, -3, 1;
1, -26, 92, -158, 145, -82, 32, -4, 1;
1, -57, 359, -906, 1265, -1135, 649, -238, 67, -5, 1;
1, -120, 1311, -4798, 9630, -12132, 10163, -5970, 2406, -620, 135, -6, 1;
MATHEMATICA
p[x_, n_]:= ((1+x-x^2)^(n+1))*Sum[(j+1)^n*(-x+x^2)^j, {j, 0, Infinity}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten
PROG
(Sage)
def T(n, k): return ( (1+x-x^2)^(n+1)*sum((j+1)^n*(x^2-x)^j for j in (0..2*n+1)) ).series(x, 2*n+3).list()[k]
[1]+flatten([[T(n, k) for k in (0..2*n-2)] for n in (0..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