A022169 -id:A022169 - cp's OEIS frontend

A015118 Triangle of q-binomial coefficients for q=-8.

1, 1, 1, 1, -7, 1, 1, 57, 57, 1, 1, -455, 3705, -455, 1, 1, 3641, 236665, 236665, 3641, 1, 1, -29127, 15150201, -120935815, 15150201, -29127, 1, 1, 233017, 969583737, 61934287481, 61934287481, 969583737, 233017, 1, 1, -1864135, 62053592185

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), A014990 (k=1), A015259 (k=2), A015276 (k=3), A015294 (k=4), A015313 (k=5), A015331 (k=6), A015347 (k=7), A015364 (k=8), A015380 (k=9), A015394 (k=10), A015413 (k=11), A015431 (k=12). - M. F. Hasler, Nov 04 2012

Index entries related to Gaussian binomial coefficients.

Cf. analog triangles for negative q=-2,...,-15: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015121 (q=-9), 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, -8], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)

T015118(n, k, q=-8)=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

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

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

A006119 Sum of Gaussian binomial coefficients [ n,k ] for q=5.

Original entry on oeis.org

1, 2, 8, 64, 1120, 42176, 3583232, 666124288, 281268665344, 260766671206400, 549874114073747456, 2547649010961476288512, 26854416724405008878829568

Offset: 0

N. J. A. Sloane

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Vincenzo Librandi, Table of n, a(n) for n = 0..75
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

Row sums of triangle A022169.

[n le 2 select n else 2*Self(n-1)+(5^(n-2)-1)*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 13 2016

Flatten[{1,RecurrenceTable[{a[n]==2*a[n-1]+(5^(n-1)-1)*a[n-2],a[0]==1,a[1]==2},a,{n,1,15}]}] (* Vaclav Kotesovec, Aug 21 2013 *)
Table[Sum[QBinomial[n, k, 5], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2016 *)

a(n) = 2*a(n-1)+(5^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - Vaclav Kotesovec, Aug 21 2013

a(n) ~ c * 5^(n^2/4), where c = EllipticTheta[3,0,1/5]/QPochhammer[1/5,1/5] = 1.845509008203... if n is even and c = EllipticTheta[2,0,1/5]/QPochhammer[1/5,1/5] = 1.829548121746... if n is odd. - Vaclav Kotesovec, Aug 21 2013

A347488 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 = 5.

Original entry on oeis.org

1, 1, 6, 1, 31, 186, 1, 156, 806, 4836, 29016, 1, 781, 20306, 121836, 629486, 3776916, 22661496, 1, 3906, 508431, 2558556, 3050586, 79315236, 409795386, 475891416, 2458772316, 14752633896, 88515803376, 1, 19531, 12714681, 320327931, 76288086

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_5)^n with dimension increments (e_1,...,e_r).

			The number of subspace chains 0 < V_1 < V_2 < (F_5)^3 is 186 = T(3, (1, 1, 1)). There are 31 = A022169(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 6 = A022169(2, 1) extensions to a two-dimensional subspace V_2.
Triangle begins:
  k:  1   2    3     4      5
      -----------------------
n=1:  1
n=2:  1   6
n=3:  1  31  186
n=4:  1 156  806  4836  29016

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

Álvar Ibeas, First 20 rows, flattened

Cf. A036038 (q = 1), A022169, A015004 (last entry in each row).

T(n, (n)) = 1. T(n, L) = A022169(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.

A347972 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_5)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 19, 56, 19, 1, 1, 33, 289, 289, 33, 1, 1, 55, 1358, 4836, 1358, 55, 1, 1, 85, 5771, 80605, 80605, 5771, 85, 1, 1, 128, 22594, 1271870, 5525686, 1271870, 22594, 128, 1, 1, 183, 81802, 18478460, 372302962, 372302962, 18478460, 81802, 183, 1

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    6
      -------------------------------
n=0:  1
n=1:  1    1
n=2:  1    4    1
n=3:  1    9    9    1
n=4:  1   19   56   19    1
n=5:  1   33  289  289   33    1
n=6:  1   55 1358 4836 1358   55    1
There are 6 = A022169(2, 1) one-dimensional subspaces in (F_5)^2. By coordinate swap, <(0, 1)> is identified with <(1, 0)> and <(1, 2)> with <(1, 3)>, while <(1, 1)> and <(1, 4)> rest invariant. Hence, T(2, 1) = 4.

Álvar Ibeas, Entries up to T(12, 5)
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
Álvar Ibeas, Column k=5 up to n=100

Cf. A022169, A008610, A241926.

T(n, 1) = T(n - 1, 1) + A008610(n).

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

Roger L. Bagula, Feb 18 2009

			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;

G. C. Greubel, Antidiagonal rows n = 0..25, flattened

Cf. A000984, A022166, A022167, A022168, A022169, A022170, A022171, A022175.

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

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 *)

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

T(n, k) = q-binomial(2*n, n, k+1), where q-binomial(n, k, q) = Product_{j=0..k-1} ( (1-q^(n-j))/(1-q^(j+1)) ), read by antidiagonals. - G. C. Greubel, Jun 14 2021

Edited by G. C. Greubel, Jun 14 2021

cp's OEIS Frontend

A015118 Triangle of q-binomial coefficients for q=-8.

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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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A006119 Sum of Gaussian binomial coefficients [ n,k ] for q=5.

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A347488 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 = 5.

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A347972 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_5)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).