A022172 -id:A022172 - 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

A347490 Irregular triangle read by rows: T(n, k) is the q-multinomial coefficient defined by the k-th partition of n in Abramowitz-Stegun order, evaluated at q = 8.

Original entry on oeis.org

1, 1, 9, 1, 73, 657, 1, 585, 4745, 42705, 384345, 1, 4681, 304265, 2738385, 22211345, 199902105, 1799118945, 1, 37449, 19477641, 156087945, 175298769, 11394419985, 92421406545, 102549779865, 831792658905, 7486133930145, 67375205371305, 1, 299593

Offset: 1

Álvar Ibeas, Sep 03 2021

Abuse of notation: we write T(n, L) for T(n, k), where L is the k-th partition of n in A-St order.

For any permutation (e_1,...,e_r) of the parts of L, T(n, L) is the number of chains of subspaces 0 < V_1 < ··· < V_r = (F_8)^n with dimension increments (e_1,...,e_r).

			The number of subspace chains 0 < V_1 < V_2 < (F_8)^3 is 657 = T(3, (1, 1, 1)). There are 73 = A022172(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 9 = A022172(2, 1) extensions to a two-dimensional subspace V_2.
Triangle begins:
  k:  1   2    3     4      5
      -----------------------
n=1:  1
n=2:  1   9
n=3:  1  73  657
n=4:  1 585 4745 42705 384345

R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.

Álvar Ibeas, First 20 rows, flattened

Cf. A036038 (q = 1), A022172, A015007 (last entry in each row).

T(n, (n)) = 1. T(n, L) = A022172(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.

A347974 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_8)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 17, 17, 1, 1, 47, 242, 47, 1, 1, 113, 3071, 3071, 113, 1, 1, 245, 34477, 232290, 34477, 245, 1, 1, 491, 341633, 16665755, 16665755, 341633, 491, 1, 1, 920, 3022045, 1073874283, 8241549097, 1073874283, 3022045, 920, 1, 1, 1635, 24145695

Offset: 0

Álvar Ibeas, Sep 21 2021

Columns can be computed by a method analogous to that of Fripertinger for isometry classes of linear codes, disallowing scalar transformation of individual coordinates.

Regarding the formula for column k = 1, note that A241926(q - 1, n) counts, up to coordinate permutation, one-dimensional subspaces of (F_q)^n generated by a vector with no zero component.

			Triangle begins:
  k:  0    1    2    3    4    5
      --------------------------
n=0:  1
n=1:  1    1
n=2:  1    5    1
n=3:  1   17   17    1
n=4:  1   47  242   47    1
n=5:  1  113 3071 3071  113    1
There are 9 = A022172(2, 1) one-dimensional subspaces in (F_8)^2. Among them, <(1, 1)> is invariant by coordinate swap and the rest are grouped in orbits of size two. Hence, T(2, 1) = 5.

Álvar Ibeas, Entries up to T(10, 4)
H. Fripertinger, Isometry classes of codes
Álvar Ibeas, Column k=1 up to n=100
Álvar Ibeas, Column k=2 up to n=100
Álvar Ibeas, Column k=3 up to n=100
Álvar Ibeas, Column k=4 up to n=100

Cf. A022172, A032192, A241926.

T(n, 1) = T(n - 1, 1) + A032192(n + 7).

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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A347490 Irregular triangle read by rows: T(n, k) is the q-multinomial coefficient defined by the k-th partition of n in Abramowitz-Stegun order, evaluated at q = 8.

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A347974 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_8)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

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