A015109 Triangle of Gaussian (or q-binomial) coefficients for q = -2.

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, 1, -341, 58311

Offset: 0

Author

Olivier Gérard

Keywords

• sign
• tabl
• easy

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

Examples

			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)
		

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.

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).

Analogous triangles for other q: A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015118 (q=-8), A015121 (q=-9), A015123 (q=-10), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15).

Programs

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

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 *)

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

Sage

flatten([[q_binomial(n,k,-2) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Nov 30 2021

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

Extensions

Edited by M. F. Hasler, Nov 04 2012