A015121
Triangle of q-binomial coefficients for q=-9.
Original entry on oeis.org
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
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
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Table[QBinomial[n, k, -9], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)
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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
A015123
Triangle of q-binomial coefficients for q=-10.
Original entry on oeis.org
1, 1, 1, 1, -9, 1, 1, 91, 91, 1, 1, -909, 9191, -909, 1, 1, 9091, 918191, 918191, 9091, 1, 1, -90909, 91828191, -917272809, 91828191, -90909, 1, 1, 909091, 9182728191, 917364637191, 917364637191, 9182728191, 909091, 1, 1, -9090909, 918273728191
Offset: 0
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),
A015121 (q=-9),
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
-
Table[QBinomial[n, k, -10], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)
-
T015123(n, k, q=-10)=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
A015124
Triangle of q-binomial coefficients for q=-11.
Original entry on oeis.org
1, 1, 1, 1, -10, 1, 1, 111, 111, 1, 1, -1220, 13542, -1220, 1, 1, 13421, 1637362, 1637362, 13421, 1, 1, -147630, 198134223, -2177691460, 198134223, -147630, 1, 1, 1623931, 23974093353, 2898705467483, 2898705467483, 23974093353, 1623931, 1, 1
Offset: 0
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),
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
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T015124(n, k, q=-11)=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
A015125
Triangle of q-binomial coefficients for q=-12.
Original entry on oeis.org
1, 1, 1, 1, -11, 1, 1, 133, 133, 1, 1, -1595, 19285, -1595, 1, 1, 19141, 2775445, 2775445, 19141, 1, 1, -229691, 399683221, -4793193515, 399683221, -229691, 1, 1, 2756293, 57554154133, 8283038077141, 8283038077141, 57554154133, 2756293, 1, 1
Offset: 0
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),
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
-
T015125(n, k, q=-12)=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
A015132
Triangle of (Gaussian) q-binomial coefficients for q=-14.
Original entry on oeis.org
1, 1, 1, 1, -13, 1, 1, 183, 183, 1, 1, -2561, 36051, -2561, 1, 1, 35855, 7063435, 7063435, 35855, 1, 1, -501969, 1384469115, -19375002205, 1384469115, -501969, 1, 1, 7027567, 271355444571, 53166390519635, 53166390519635, 271355444571
Offset: 0
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),
A015129 (q=-13),
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
-
T015132(n, k, q=-14)=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
A156914
Square array T(n, k) = q-binomial(2*n, n, k+1), read by antidiagonals.
Original entry on oeis.org
1, 1, 2, 1, 3, 6, 1, 4, 35, 20, 1, 5, 130, 1395, 70, 1, 6, 357, 33880, 200787, 252, 1, 7, 806, 376805, 75913222, 109221651, 924, 1, 8, 1591, 2558556, 6221613541, 1506472167928, 230674393235, 3432, 1, 9, 2850, 12485095, 200525284806, 1634141006295525, 267598665689058580, 1919209135381395, 12870
Offset: 0
Square array begins as:
1, 1, 1, 1, ...;
2, 3, 4, 5, ...;
6, 35, 130, 357, ...;
20, 1395, 33880, 376805, ...;
70, 200787, 75913222, 6221613541, ...;
252, 109221651, 1506472167928, 1634141006295525, ...;
Antidiagonal triangle begins as:
1;
1, 2;
1, 3, 6;
1, 4, 35, 20;
1, 5, 130, 1395, 70;
1, 6, 357, 33880, 200787, 252;
1, 7, 806, 376805, 75913222, 109221651, 924;
1, 8, 1591, 2558556, 6221613541, 1506472167928, 230674393235, 3432;
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QBinomial:= func< n,k,q | q eq 1 select Binomial(n, k) else k eq 0 select 1 else (&*[ (1-q^(n-j+1))/(1-q^j): j in [1..k] ]) >;
T:= func< n,k | QBinomial(2*n, n, k+1) >;
[T(k, n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 14 2021
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T[n_, k_]:= QBinomial[2*n, n, k+1];
Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 14 2021 *)
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def A156914(n, k): return q_binomial(2*n, n, k+1)
flatten([[A156914(k,n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 14 2021
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