A156281 Irregular table with the coefficient [x^k] of Product_{j=1..n} (x^j - (1-x^j)/(1-x)) in row n, column 0<=k.

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, 1, 4, 8, 11, 10, 5, -5, -15, -19, -17, -7, 5, 13, 9, 7, -1, -7, -8, 1, 10, -6, 1, -1, -5, -13, -24, -34, -39, -34, -17, 9, 38, 59, 63, 50, 26, -6, -28, -36, -25, -9, 2, 13, 17, 8, -5, -14, -4, 15, -7, 1

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

Sequence in context: A325615 A029434 A358192 * A002217 A344173 A157047

Adjacent sequences: A156278 A156279 A156280 * A156282 A156283 A156284

Keyword

tabf,sign

Author

Roger L. Bagula, Feb 07 2009

Extensions

Edited by G. C. Greubel, Jan 03 2022

Status

approved