A022180 -id:A022180 - cp's OEIS frontend

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

Olivier Gérard

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

Index entries related to Gaussian binomial coefficients.

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

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

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

Olivier Gérard

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 in the former, or row/columns in the latter, are then (k=0,...,12): A000012, A014992, A015261, A015278, A015298, A015316, A015333, A015350, A015367, A015382, A015398, A015417, A015433. - M. F. Hasler, Nov 04 & Nov 05 2012

Index entries related to Gaussian binomial coefficients.

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

Olivier Gérard

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 in the former, or row/columns in the latter, are then (k=0,...,12): A000012, A014993, A015262, A015279, A015300, A015317, A015334, A015353, A015368, A015383, A015499, A015418, A015434. - M. F. Hasler, Nov 04 & Nov 05 2012

Index entries related to Gaussian binomial coefficients.

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

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

Olivier Gérard

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, for k=0,...,12: A000012, A014994, A015264, A015281, A015302, A015319, A015336, A015354, A015369, A015384, A015401, A015421, A015436. - M. F. Hasler, Nov 04 2012

Index entries related to Gaussian binomial coefficients.

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

Olivier Gérard

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). - M. F. Hasler, Nov 04 2012

Index entries related to Gaussian binomial coefficients.

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

A015204 Sum of Gaussian binomial coefficients for q=16.

Original entry on oeis.org

1, 2, 19, 548, 78901, 36070982, 82805758039, 605336231791208, 22229212008282455161, 2599943776445794193452682, 1527585017328101696333407084699, 2858671468933430533899194300073611948, 26873565540839814480301520088779437828129981

Offset: 0

N. J. A. Sloane, Olivier Gérard

nonn

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Row sums of triangle A022180.

Total/@Table[QBinomial[n, m, 16], {n, 0, 20}, {m, 0, n}] (* Vincenzo Librandi, Nov 02 2012 *)

a(0) = 1, a(1) = 2, a(n) = 2*a(n-1) + a(n-2)*((16^(n-1)) - 1). - Vincenzo Librandi, Nov 02 2012

A173583 Triangle T(n, k, q) = q-binomial(n, k, q^2), for q = 5, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 26, 1, 1, 651, 651, 1, 1, 16276, 407526, 16276, 1, 1, 406901, 254720026, 254720026, 406901, 1, 1, 10172526, 159200423151, 3980255126276, 159200423151, 10172526, 1, 1, 254313151, 99500274641901, 62191645548485651, 62191645548485651, 99500274641901, 254313151, 1

Offset: 0

Roger L. Bagula, Feb 22 2010

Row sums are: 1, 2, 28, 1304, 440080, 510253856, 4298676317632, 124582292154881408, ...

			Triangle begins as:
  1;
  1,        1;
  1,       26,            1;
  1,      651,          651,             1;
  1,    16276,       407526,         16276,            1;
  1,   406901,    254720026,     254720026,       406901,        1;
  1, 10172526, 159200423151, 3980255126276, 159200423151, 10172526, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000012 (q=0), A007318 (q=1), A022168 (q=2), A022173 (q=3), A022180 (q=4), A173583 (q=5).

q:=5;; [q^(k*(n-k))*GaussianBinomial(n, k, q): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 22 2021

(* First program *)
c[n_, q_]:= Product[(1 -q^(2*j))/(1-q), {j,1,n}];
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
Table[T[n, k, 5], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
Table[QBinomial[n,k,5^2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 22 2021 *)
T[n_, k_, p_]:= T[n, k, p] = If[k==0 || k==n, 1, T[n-1, k-1, p] + p^k*T[n-1, k, q]];  Table[T[n, k, 25], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 22 2021 *)

flatten([[q_binomial(n, k, 5^2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 22 2021

T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)) where c(n, q) = Product_{j=1..n} (1 -q^(2*j))/(1-q) for q = 5.
From G. C. Greubel, Feb 22 2021: (Start)
T(n, k, q) = q-binomial(n, k, q^2), for q = 5.
T(n, k) = T(n-1, k-1) + p^k * T(n-1, k), with p = 25 (as a number triangle). (End)

Edited by G. C. Greubel, Feb 22 2021

cp's OEIS Frontend

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

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A015123 Triangle of q-binomial coefficients for q=-10.

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A015124 Triangle of q-binomial coefficients for q=-11.

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A015125 Triangle of q-binomial coefficients for q=-12.

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A015132 Triangle of (Gaussian) q-binomial coefficients for q=-14.

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A015204 Sum of Gaussian binomial coefficients for q=16.

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A173583 Triangle T(n, k, q) = q-binomial(n, k, q^2), for q = 5, read by rows.

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