%I #27 Sep 08 2022 08:44:39

%S 1,1,1,1,-12,1,1,157,157,1,1,-2040,26690,-2040,1,1,26521,4508570,

%T 4508570,26521,1,1,-344772,761974851,-9900819720,761974851,-344772,1,

%U 1,4482037,128773405047,21752862899691,21752862899691,128773405047,4482037,1

%N Triangle of (Gaussian) q-binomial coefficients for q = -13.

%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, resp. rows (or columns) of the latter, are: A000012 (all 1's), A015000 (q-integers for q=-13), A015265 (k=2), A015286 (k=3), A015303 (k=4), A015321 (k=5), A015337 (k=6), A015355 (k=7), A015370 (k=8), A015385 (k=9), A015402 (k=10), A015422 (k=11), A015438 (k=12). - _M. F. Hasler_, Nov 04 2012

%H G. C. Greubel, <a href="/A015129/b015129.txt">Rows n = 0..50 of the triangle, flattened</a>

%H <a href="/index/Ga#Gaussian_binomial_coefficients">Index entries related to Gaussian binomial coefficients</a>.

%F As a triangle, T(n, k) = Product_{i=1..k} ((-13)^(1+n-i)-1)/((-13)^i-1), with 0 <= k <= n = 0,1,2,...

%e The square array looks as follows:

%e 1 1 1 1 1 1 ...

%e 1 -12 157 -2040 26521 -344772 ...

%e 1 157 26690 4508570 761974851 128773405047 ...

%e 1 -2040 4508570 -9900819720 21752862899691 ...

%e 1 26521 761974851 21752862899691 621305270140974342 ...

%e 1 -344772 128773405047 -47790911017216080 17745052029585350965782 ...

%e (...)

%t Flatten[Table[QBinomial[x,y,-13],{x,0,10},{y,0,x}]] (* _Harvey P. Dale_, Jul 12 2014 *)

%o (PARI) A015129(n, r, q=-13)=prod(i=1, r, (q^(1+n-i+r)-1)/(q^i-1)) \\ (Indexing is that of the square array: n,r=0,1,2,...) - _M. F. Hasler_, Nov 03 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,-13): k in [0..n], n in [0..10]]; // A015129 // _G. C. Greubel_, Dec 01 2021

%o (Sage) flatten([[q_binomial(n,k,-13) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Dec 01 2021

%Y Cf. analog triangles for other negative q=-2,...,-15: A015109 (q=-2), 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), A015132 (q=-14), A015133 (q=-15). - _M. F. Hasler_, Nov 04 2012

%Y Cf. analog triangles for positive q=2,...,24: 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). - _M. F. Hasler_, Nov 05 2012

%K sign,tabl,easy

%O 0,5

%A _Olivier Gérard_