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Triangle of q-binomial coefficients for q=-3.
14

%I #13 Nov 04 2012 10:44:17

%S 1,1,1,1,-2,1,1,7,7,1,1,-20,70,-20,1,1,61,610,610,61,1,1,-182,5551,

%T -15860,5551,-182,1,1,547,49777,433771,433771,49777,547,1,1,-1640,

%U 448540,-11662040,35569222,-11662040,448540,-1640,1,1,4921,4035220,315323620

%N Triangle of q-binomial coefficients for q=-3.

%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), A014983 (k=1), A015251 (k=2), A015268 (k=3), A015288 (k=4), A015306 (k=5), A015324 (k=6), A015340 (k=7), A015357 (k=8), A015375 (k=9), A015388 (k=10), A015407 (k=11), A015424 (k=12),... - _M. F. Hasler_, Nov 04 2012

%H Vincenzo Librandi, <a href="/A015110/b015110.txt">Table of n, a(n) for n = 0..1000</a>

%t Flatten[Table[QBinomial[n, m, -3], {n, 0, 50}, {m, 0, n}]] (* _Vincenzo Librandi_, Nov 01 2012 *)

%o (PARI) T015110(n, k, q=-3)=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

%Y Cf. analog triangles for other q: A015109 (q=-2), 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); A022166 (q=2), A022167 (q=3), A022168, A022169, A022170, A022171, A022172, A022173, A022174 (q=10), A022175, A022176, A022177, A022178, A022179, A022180, A022181, A022182, A022183, A022184 (q=20), A022185, A022186, A022187, A022188. - _M. F. Hasler_, Nov 04 2012

%K sign,tabl,easy

%O 0,5

%A _Olivier GĂ©rard_