OFFSET
0,6
LINKS
G. C. Greubel, Rows n = 0..25 of the irregular triangle, flattened
L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. Volume 15, Number 4 (1948), 987-1000.
L. Carlitz, J. Riordan, Two element lattice permutation numbers and their q-generalization, Duke Math. J. Volume 31, Number 3 (1964), 371-388.
John Shareshian and Michelle L. Wachs, q-Eulerian Polynomials: Excedance Number and Major Index, arXiv:math/0608274 [math.CO], 2006, page 3.
FORMULA
T(n, k) = [x^k]( Product_{j=1..n} (x^j - (1-x^j)/(1-x)) ).
Sum_{k=0..binomial(n+1,2)} T(n, k) = A000007(n).
EXAMPLE
Irregular triangle begins as:
1;
-1, 1;
1, 0, -2, 1;
-1, -1, 1, 2, 1, -3, 1;
1, 2, 1, -1, -4, -2, 0, 3, 3, -4, 1;
-1, -3, -4, -3, 1, 5, 8, 5, -1, -4, -5, -3, 3, 6, -5, 1;
MATHEMATICA
T[n_]:= CoefficientList[(-1)^n*(1-x)^(-n)*Product[1 -2*x^j +x^(j+1), {j, n}], x];
Table[T[n], {n, 0, 10}] //Flatten (* modified by G. C. Greubel, Jan 03 2022 *)
PROG
(Sage)
def T(n, k): return ( (-1)^n*(1-x)^(-n)*product( 1 -2*x^j +x^(j+1) for j in (1..n)) ).series(x, 1+binomial(n+1, 2)).list()[k]
flatten([[T(n, k) for k in (0..binomial(n+1, 2))] for n in (0..10)]) # G. C. Greubel, Jan 03 2022
CROSSREFS
KEYWORD
tabf,sign
AUTHOR
Roger L. Bagula, Feb 07 2009
EXTENSIONS
Edited by G. C. Greubel, Jan 03 2022
STATUS
approved