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A015129 Triangle of (Gaussian) q-binomial coefficients for q=-13. 16
1, 1, 1, 1, -12, 1, 1, 157, 157, 1, 1, -2040, 26690, -2040, 1, 1, 26521, 4508570, 4508570, 26521, 1, 1, -344772, 761974851, -9900819720, 761974851, -344772, 1, 1, 4482037, 128773405047, 21752862899691, 21752862899691, 128773405047, 4482037, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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, 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

LINKS

Table of n, a(n) for n=0..36.

Index entries related to Gaussian binomial coefficients.

FORMULA

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,...

EXAMPLE

The square array looks as follows:

1    1          1              1                      1               1       ...

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

1   157       26690         4508570               761974851      128773405047 ...

1  -2040     4508570      -9900819720           21752862899691        ...

1  26521    761974851    21752862899691       621305270140974342      ...

1 -344772 128773405047 -47790911017216080  17745052029585350965782    ...

(...)

MATHEMATICA

Flatten[Table[QBinomial[x, y, -13], {x, 0, 10}, {y, 0, x}]] (* Harvey P. Dale, Jul 12 2014 *)

PROG

(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

CROSSREFS

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

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

Sequence in context: A166962 A022175 A176627 * A172376 A289673 A010211

Adjacent sequences:  A015126 A015127 A015128 * A015130 A015131 A015132

KEYWORD

sign,tabl,easy

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified December 14 19:27 EST 2019. Contains 329987 sequences. (Running on oeis4.)