OFFSET
0,8
COMMENTS
May be read as a symmetric triangular (T(n,k)=T(n,n-k); k=0,...,n; n=0,1,...) or square array (A(n,r)=A(r,n)=T(n+r,r), read by antidiagonals). The diagonals of the former (or rows/columns of the latter) are A000012 (k=0), A077925 (k=1), A015249 (k=2), A015266 (k=3), A015287 (k=4), A015305 (k=5), A015323 (k=6), A015338 (k=7), A015356 (k=8), A015371 (k=9), A015386 (k=10), A015405 (k=11), A015423 (k=12), ... - M. F. Hasler, Nov 04 2012
The elements of the inverse matrix are apparently T^(-1)(n,k) = (-1)^n*A157785(n,k). - R. J. Mathar, Mar 12 2013
Fu et al. give two combinatorial interpretations of the (unsigned) q-binomial coefficients when q is a negative integer. - Peter Bala, Nov 02 2017
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
J. A. de Azcarraga and J. A. Macfarlane, Group Theoretical Foundations of Fractional Supersymmetry, arxiv:hep-th/9506177 (1995).
S. Fu, V. Reiner, D. Stanton and N. Thiem, The negative q-binomial, arXiv:1108.4702 [math.CO], 2011.
R. Parthasarathy, q-Fermionic Numbers and Their Roles in Some Physical Problems, arxiv:quant-ph/0403216, 2004.
FORMULA
T(n, k) = q-binomial(n, k, -2).
T(n, k, q) = Product_{j=1..k} ( (1 - q^(n-j+1))/(1 - q^j) ), for q = -2. - Roger L. Bagula, Feb 10 2009
EXAMPLE
From Roger L. Bagula, Feb 10 2009: (Start)
1;
1, 1;
1, -1, 1;
1, 3, 3, 1;
1, -5, 15, -5, 1;
1, 11, 55, 55, 11, 1;
1, -21, 231, -385, 231, -21, 1;
1, 43, 903, 3311, 3311, 903, 43, 1;
1, -85, 3655, -25585, 56287, -25585, 3655, -85, 1;
1, 171, 14535, 208335, 875007, 875007, 208335, 14535, 171, 1; (End)
MAPLE
A015109 := proc(n, k)
mul( ((-2)^(1+n-i)-1)/((-2)^i-1) , i=1..k) ;
end proc: # R. J. Mathar, Mar 12 2013
MATHEMATICA
T[n_, k_, q_]:= Product[(1 - q^(n-j+1))/(1 - q^j), {j, k}];
Table[T[n, k, -2], {n, 0, 10}, {k, 0, n}]//Flatten (* Roger L. Bagula, Feb 10 2009 *)(* modified by G. C. Greubel, Nov 30 2021 *)
Table[QBinomial[n, k, -2], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Apr 09 2016 *)
PROG
(PARI) T015109(n, k, q=-2)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0, 1, 2, ...) \\ M. F. Hasler, Nov 04 2012
(Magma)
qBinomial:= func< n, k, q | k eq 0 select 1 else (&*[(1-q^(n-j+1))/(1-q^j): j in [1..k]]) >;
[qBinomial(n, k, -2): k in [0..n], n in [0..10]]; // A015109 // G. C. Greubel, Nov 30 2021
(Sage) flatten([[q_binomial(n, k, -2) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Nov 30 2021
CROSSREFS
Cf. A015152 (row sums).
Cf. A022166 (q=2), A022167 (q=3), A022168 (q=4), A022169 (q=5), A022170 (q=6), A022171 (q=7), A022172 (q=8), A022173 (q=9), A022174 (q=10), A022175 (q=11), A022176 (q=12), A022177 (q=13), A022178 (q=14), A022179 (q=15), A022180 (q=16), A022181 (q=17), A022182 (q=18), A022183 (q=19), A022184 (q=20), A022185 (q=21), A022186 (q=22), A022187 (q=23), A022188 (q=24).
KEYWORD
AUTHOR
EXTENSIONS
Edited by M. F. Hasler, Nov 04 2012
STATUS
approved