A015121 Triangle of q-binomial coefficients for q=-9.

1, 1, 1, 1, -8, 1, 1, 73, 73, 1, 1, -656, 5986, -656, 1, 1, 5905, 484210, 484210, 5905, 1, 1, -53144, 39226915, -352504880, 39226915, -53144, 1, 1, 478297, 3177326971, 257015284435, 257015284435, 3177326971, 478297, 1, 1, -4304672, 257363962948

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), A014991 (k=1), A015260 (k=2), A015277 (k=3), A015295 (k=4), A015315 (k=5), A015332 (k=6), A015349 (k=7), A015365 (k=8), A015381 (k=9), A015397 (k=10), A015414 (k=11), A015432 (k=12). - M. F. Hasler, Nov 05 2012

Links

• Index entries related to Gaussian binomial coefficients.

Crossrefs

Cf. analog triangles for other q: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015118 (q=-8), A015123 (q=-10), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15). - M. F. Hasler, Nov 04 2012

Cf. analog triangles for positive q=2,...,24: 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 05 2012

Programs

Mathematica

Table[QBinomial[n, k, -9], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)

PARI

T015121(n, k, q=-9)=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