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Triangle of Gaussian (or q-binomial) coefficients for q = -2.
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%I #57 Sep 08 2022 08:44:39

%S 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,

%T -21,1,1,43,903,3311,3311,903,43,1,1,-85,3655,-25585,56287,-25585,

%U 3655,-85,1,1,171,14535,208335,875007,875007,208335,14535,171,1,1,-341,58311

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

%C 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

%C The elements of the inverse matrix are apparently T^(-1)(n,k) = (-1)^n*A157785(n,k). - _R. J. Mathar_, Mar 12 2013

%C Fu et al. give two combinatorial interpretations of the (unsigned) q-binomial coefficients when q is a negative integer. - _Peter Bala_, Nov 02 2017

%H Vincenzo Librandi, <a href="/A015109/b015109.txt">Table of n, a(n) for n = 0..1000</a>

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

%H J. A. de Azcarraga and J. A. Macfarlane, <a href="http://arxiv.org/abs/hep-th/9506177">Group Theoretical Foundations of Fractional Supersymmetry</a>, arxiv:hep-th/9506177 (1995).

%H S. Fu, V. Reiner, D. Stanton and N. Thiem, <a href="https://arxiv.org/abs/1108.4702">The negative q-binomial</a>, arXiv:1108.4702 [math.CO], 2011.

%H R. Parthasarathy, <a href="http://arxiv.org/abs/quant-ph/0403216">q-Fermionic Numbers and Their Roles in Some Physical Problems</a>, arxiv:quant-ph/0403216, 2004.

%F T(n, k) = q-binomial(n, k, -2).

%F 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

%e From _Roger L. Bagula_, Feb 10 2009: (Start)

%e 1;

%e 1, 1;

%e 1, -1, 1;

%e 1, 3, 3, 1;

%e 1, -5, 15, -5, 1;

%e 1, 11, 55, 55, 11, 1;

%e 1, -21, 231, -385, 231, -21, 1;

%e 1, 43, 903, 3311, 3311, 903, 43, 1;

%e 1, -85, 3655, -25585, 56287, -25585, 3655, -85, 1;

%e 1, 171, 14535, 208335, 875007, 875007, 208335, 14535, 171, 1; (End)

%p A015109 := proc(n, k)

%p mul( ((-2)^(1+n-i)-1)/((-2)^i-1) ,i=1..k) ;

%p end proc: # _R. J. Mathar_, Mar 12 2013

%t T[n_, k_, q_]:= Product[(1 - q^(n-j+1))/(1 - q^j), {j, k}];

%t 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 *)

%t Table[QBinomial[n, k, -2], {n,0,10}, {k,0,n}]//Flatten (* _Jean-François Alcover_, Apr 09 2016 *)

%o (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

%o (Magma)

%o qBinomial:= func< n,k,q | k eq 0 select 1 else (&*[(1-q^(n-j+1))/(1-q^j): j in [1..k]]) >;

%o [qBinomial(n,k,-2): k in [0..n], n in [0..10]]; // A015109 // _G. C. Greubel_, Nov 30 2021

%o (Sage) flatten([[q_binomial(n,k,-2) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Nov 30 2021

%Y Cf. A015152 (row sums).

%Y 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).

%Y 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).

%K sign,tabl,easy

%O 0,8

%A _Olivier Gérard_

%E Edited by _M. F. Hasler_, Nov 04 2012