A022167 -id:A022167 - cp's OEIS frontend

A125813 q-Bell numbers for q=3; eigensequence of A022167, which is the triangle of Gaussian binomial coefficients [n,k] for q=3.

1, 1, 2, 7, 47, 628, 17327, 1022983, 132615812, 38522717107, 25526768401271, 39190247441314450, 141213238745969102393, 1207367655155905204747681, 24733467452839301566047854678, 1224709126636123500201799360630423, 147747406166666863538672620806542995763

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

			The recurrence: a(n) = Sum_{k=0..n-1} A022167(n-1,k) * a(k) is illustrated by:
  a(2) = 1*(1) + 4*(1) + 1*(2) = 7;
  a(3) = 1*(1) + 13*(1) + 13*(2) + 1*(7) = 47;
  a(4) = 1*(1) + 40*(1) + 130*(2) + 40*(7) + 1*(47) = 628.
Triangle A022167 begins:
  1;
  1, 1;
  1, 4, 1;
  1, 13, 13, 1;
  1, 40, 130, 40, 1;
  1, 121, 1210, 1210, 121, 1;
  1, 364, 11011, 33880, 11011, 364, 1;
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..72

Cf. A022167, A125810, A125811, A125812, A125814, A125815.

Column k=3 of A381369.

b:= proc(o, u, t) option remember;
     `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2), 0)+add(3^(u+j-1)*
        b(o-j, u+j-1, min(2, t+1)), j=`if`(t=0, 1, 1..o)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..18);  # Alois P. Heinz, Feb 21 2025

b[o_, u_, t_] := b[o, u, t] =
   If[u + o == 0, 1, If[t > 0, b[u + o, 0, 0], 0] + Sum[3^(u + j - 1)*
   b[o - j, u + j - 1, Min[2, t + 1]], {j, If[t == 0, {1}, Range[o]]}]];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Mar 15 2025, after Alois P. Heinz *)

/* q-Binomial coefficients: */
C_q(n,k)=if(n

a(n) = Sum_{k=0..n-1} A022167(n-1,k) * a(k) for n>0, with a(0)=1.

a(n) = Sum_{k>=0} 3^k * A125810(n,k). - Alois P. Heinz, Feb 21 2025

A143777 Eigentriangle of triangle A022167.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 13, 26, 7, 1, 40, 260, 280, 47, 1, 121, 2420, 8470, 5687, 628

Offset: 0

Gary W. Adamson, Aug 31 2008

Row sums of the triangle = A125813 shifted one place to the left = (1, 2, 7, 47, 628,...).
Row sums of row n terms = rightmost term of row (n+1).
Example: rightmost term of row 3 = 7 = (1 + 4 + 2).
Triangle A022167 =
1;
1, 1;
1, 4, 1;
1, 13, 13, 1;
1, 40, 130, 40, 1;
... The eigensequence of A022167 = A125815: (1, 1, 2, 7, 47, 628, 17327,...).
Triangle A143777 applies a termwise product of the first n terms of (1, 1, 2, 7, 47,...) and the (n-1)-th row terms of triangle A022167.

			First few rows of the triangle are:
  1;
  1, 1;
  1, 4, 2;
  1, 13, 26, 7;
  1, 40, 260, 280, 47;
  1, 121, 2420, 8470, 5687, 628;
  ...
Row 3 = (1, 13, 26, 7) = termwise product of (1, 13, 13, 1) and (1, 1, 2, 7); where (1, 13, 13, 1) = row 3 of triangle A022167 and (1, 1, 2, 7) = the first 4 terms of A125813, the eigensequence of A022167.

Cf. A022167, A125813.

Triangle read by rows, A022167 * (A125813 * 0^(n-k)); 0<=k<=n

A174527 Triangle T(n,m) = 2*A022167(n,m) - binomial(n, m), 0 <= m <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 23, 23, 1, 1, 76, 254, 76, 1, 1, 237, 2410, 2410, 237, 1, 1, 722, 22007, 67740, 22007, 722, 1, 1, 2179, 198905, 1851507, 1851507, 198905, 2179, 1, 1, 6552, 1792492, 50190504, 151826374, 50190504, 1792492, 6552, 1, 1, 19673, 16139204

Offset: 0

Roger L. Bagula and Gary W. Adamson, Mar 21 2010

Row sums are 1, 2, 8, 48, 408, 5296, 113200, 4105184, 255805472, 27442457664, 5089653253824, ... = 2*A006117(n)-2^n.

			Triangle begins
  1;
  1,    1;
  1,    6,       1;
  1,   23,      23,        1;
  1,   76,     254,       76,         1;
  1,  237,    2410,     2410,       237,        1;
  1,  722,   22007,    67740,     22007,      722,       1;
  1, 2179,  198905,  1851507,   1851507,   198905,    2179,    1;
  1, 6552, 1792492, 50190504, 151826374, 50190504, 1792492, 6552, 1;

Cf. A060187.

A174527 := proc(n,k)
        2*A022167(n,k)-binomial(n,k) ;
end proc:
seq(seq(A174527(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Nov 14 2011

c[n_, q_] = Product[1 - q^i, {i, 1, n}];
t[n_, m_, q_] = 2*c[n, q]/(c[m, q]*c[n - m, q]) - Binomial[n, m];
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]

A022166 Triangle of Gaussian binomial coefficients (or q-binomial coefficients) [n,k] for q = 2.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 35, 15, 1, 1, 31, 155, 155, 31, 1, 1, 63, 651, 1395, 651, 63, 1, 1, 127, 2667, 11811, 11811, 2667, 127, 1, 1, 255, 10795, 97155, 200787, 97155, 10795, 255, 1, 1, 511, 43435, 788035, 3309747, 3309747, 788035, 43435, 511, 1

Offset: 0

N. J. A. Sloane

Also number of distinct binary linear [n,k] codes.
Row sums give A006116.
Central terms are A006098.
T(n,k) is the number of subgroups of the Abelian group (C_2)^n that have order 2^k. - Geoffrey Critzer, Mar 28 2016
T(n,k) is the number of k-subspaces of the finite vector space GF(2)^n. - Jianing Song, Jan 31 2020

			Triangle begins:
  1;
  1,   1;
  1,   3,    1;
  1,   7,    7,     1;
  1,  15,   35,    15,     1;
  1,  31,  155,   155,    31,    1;
  1,  63,  651,  1395,   651,   63,   1;
  1, 127, 2667, 11811, 11811, 2667, 127, 1;

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.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

T. D. Noe, Rows n=0..50 of triangle, flattened
Octavio A. Agustín-Aquino, Archimedes' quadrature of the parabola and minimal covers, arXiv:1602.05279 [math.CO], 2016.
J. A. de Azcarraga and J. A. Macfarlane, Group Theoretical Foundations of Fractional Supersymmetry arXiv:hep-th/9506177, 1995.
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv:1409.3820 [math.NT], 2014.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
D. Slepian, A class of binary signaling alphabets, Bell System Tech. J. 35 (1956), 203-234.
D. Slepian, Some further theory of group codes, Bell System Tech. J. 39 1960 1219-1252.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Eric W. Weisstein, q-Binomial Coefficient.
Wikipedia, q-binomial
Index entries for sequences related to binary linear codes
Index entries for sequences related to Gaussian binomial coefficients

Cf. A006516, A218449, A135950 (matrix inverse), A000225 (k=1), A006095 (k=2), A006096 (k=3), A139382.
Cf. this sequence (q=2), A022167 (q=3), A022168 (q=4), A022169 (q=5), A022170 (q=6), A022171 (q=7), A022172 (q=8), A022173 (q=9), A022174 (q=10), A022175 (q=11), A022176 (q=12), A022177 (q=13), A022178 (q=14), A022179 (q=15), A022180 (q=16), A022181 (q=17), A022182 (q=18), A022183 (q=19), A022184 (q=20), A022185 (q=21), A022186 (q=22), A022187 (q=23), A022188 (q=24).
Analogous 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), A015123 (q=-10), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15).

q:=2; [[k le 0 select 1 else (&*[(1-q^(n-j))/(1-q^(j+1)): j in [0..(k-1)]]): k in [0..n]]: n in [0..20]]; // G. C. Greubel, Nov 17 2018

A005329 := proc(n)
   mul( 2^i-1,i=1..n) ;
end proc:
A022166 := proc(n,m)
   A005329(n)/A005329(n-m)/A005329(m) ;
end proc: # R. J. Mathar, Nov 14 2011

Table[QBinomial[n, k, 2], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 08 2016 *)
(* S stands for qStirling2 *) S[n_, k_, q_] /; 1 <= k <= n := S[n - 1, k - 1, q] + Sum[q^j, {j, 0, k - 1}]*S[n - 1, k, q]; S[n_, 0, ] := KroneckerDelta[n, 0]; S[0, k, ] := KroneckerDelta[0, k]; S[, , ] = 0;
T[n_, k_] /; n >= k := Sum[Binomial[n, j]*S[n - j, n - k, q]*(q - 1)^(k - j) /. q -> 2, {j, 0, k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 08 2020, after Vladimir Kruchinin *)

T(n,k)=polcoeff(x^k/prod(j=0,k,1-2^j*x+x*O(x^n)),n) \\ Paul D. Hanna, Oct 28 2006

qp = matpascal(9,2);
for(n=1,#qp,for(k=1,n,print1(qp[n,k],", "))) \\ Gerald McGarvey, Dec 05 2009

{q=2; T(n,k) = if(k==0,1, if (k==n, 1, if (k<0 || nG. C. Greubel, May 27 2018

def T(n,k): return gaussian_binomial(n,k).subs(q=2) # Ralf Stephan, Mar 02 2014

G.f.: A(x,y) = Sum_{k>=0} y^k/Product_{j=0..k} (1 - 2^j*x). - Paul D. Hanna, Oct 28 2006
For k = 1,2,3,... the expansion of exp( Sum_{n >= 1} (2^(k*n) - 1)/(2^n - 1)*x^n/n ) gives the o.g.f. for the k-th diagonal of the triangle (k = 1 corresponds to the main diagonal). - Peter Bala, Apr 07 2015
T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017
T(m+n,k) = Sum_{i=0..k} q^((k-i)*(m-i)) * T(m,i) * T(n,k-i), q=2 (see the Sved link, page 337). - Werner Schulte, Apr 09 2019
T(n,k) = Sum_{j=0..k} qStirling2(n-j,n-k)*C(n,j) where qStirling2(n,k) is A139382. - Vladimir Kruchinin, Mar 04 2020

A015109 Triangle of Gaussian (or q-binomial) coefficients for q = -2.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, 3, 3, 1, 1, -5, 15, -5, 1, 1, 11, 55, 55, 11, 1, 1, -21, 231, -385, 231, -21, 1, 1, 43, 903, 3311, 3311, 903, 43, 1, 1, -85, 3655, -25585, 56287, -25585, 3655, -85, 1, 1, 171, 14535, 208335, 875007, 875007, 208335, 14535, 171, 1, 1, -341, 58311

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), A077925 (k=1), A015249 (k=2), A015266 (k=3), A015287 (k=4), A015305 (k=5), A015323 (k=6), A015338 (k=7), A015356 (k=8), A015371 (k=9), A015386 (k=10), A015405 (k=11), A015423 (k=12), ... - M. F. Hasler, Nov 04 2012
The elements of the inverse matrix are apparently T^(-1)(n,k) = (-1)^n*A157785(n,k). - R. J. Mathar, Mar 12 2013
Fu et al. give two combinatorial interpretations of the (unsigned) q-binomial coefficients when q is a negative integer. - Peter Bala, Nov 02 2017

			From _Roger L. Bagula_, Feb 10 2009: (Start)
  1;
  1,   1;
  1,  -1,     1;
  1,   3,     3,      1;
  1,  -5,    15,     -5,      1;
  1,  11,    55,     55,     11,      1;
  1, -21,   231,   -385,    231,    -21,      1;
  1,  43,   903,   3311,   3311,    903,     43,     1;
  1, -85,  3655, -25585,  56287, -25585,   3655,   -85,   1;
  1, 171, 14535, 208335, 875007, 875007, 208335, 14535, 171, 1;  (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
J. A. de Azcarraga and J. A. Macfarlane, Group Theoretical Foundations of Fractional Supersymmetry, arxiv:hep-th/9506177 (1995).
S. Fu, V. Reiner, D. Stanton and N. Thiem, The negative q-binomial, arXiv:1108.4702 [math.CO], 2011.
R. Parthasarathy, q-Fermionic Numbers and Their Roles in Some Physical Problems, arxiv:quant-ph/0403216, 2004.

Cf. A015152 (row sums).
Cf. A022166 (q=2), A022167 (q=3), A022168 (q=4), A022169 (q=5), A022170 (q=6), A022171 (q=7), A022172 (q=8), A022173 (q=9), A022174 (q=10), A022175 (q=11), A022176 (q=12), A022177 (q=13), A022178 (q=14), A022179 (q=15), A022180 (q=16), A022181 (q=17), A022182 (q=18), A022183 (q=19), A022184 (q=20), A022185 (q=21), A022186 (q=22), A022187 (q=23), A022188 (q=24).
Analogous triangles for other q: 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), A015132 (q=-14), A015133 (q=-15).

qBinomial:= func< n,k,q | k eq 0 select 1 else (&*[(1-q^(n-j+1))/(1-q^j): j in [1..k]]) >;
[qBinomial(n,k,-2): k in [0..n], n in [0..10]]; // A015109 // G. C. Greubel, Nov 30 2021

A015109 := proc(n, k)
   mul( ((-2)^(1+n-i)-1)/((-2)^i-1) ,i=1..k) ;
end proc: # R. J. Mathar, Mar 12 2013

T[n_, k_, q_]:= Product[(1 - q^(n-j+1))/(1 - q^j), {j, k}];
Table[T[n,k,-2], {n,0,10}, {k,0,n}]//Flatten (* Roger L. Bagula, Feb 10 2009 *)(* modified by G. C. Greubel, Nov 30 2021 *)
Table[QBinomial[n, k, -2], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, Apr 09 2016 *)

T015109(n, k, q=-2)=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

flatten([[q_binomial(n,k,-2) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Nov 30 2021

T(n, k) = q-binomial(n, k, -2).

T(n, k, q) = Product_{j=1..k} ( (1 - q^(n-j+1))/(1 - q^j) ), for q = -2. - Roger L. Bagula, Feb 10 2009

Edited by M. F. Hasler, Nov 04 2012

A015129 Triangle of (Gaussian) q-binomial coefficients for q = -13.

Original entry on oeis.org

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

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, 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

			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    ...
(...)

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
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), 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 (q=4), A022169 (q=5), A022170 (q=6), A022171 (q=7), A022172 (q=8), A022173 (q=9), A022174 (q=10), A022175 (q=11), A022176 (q=12), A022177 (q=13), A022178 (q=14), A022179 (q=15), A022180 (q=16), A022181 (q=17), A022182 (q=18), A022183 (q=19), A022184 (q=20), A022185 (q=21), A022186 (q=22), A022187 (q=23), A022188 (q=24). - M. F. Hasler, Nov 05 2012

qBinomial:= func< n,k,q | k eq 0 select 1 else (&*[(1 -q^(n-j+1))/(1 -q^j): j in [1..k]]) >;
[qBinomial(n,k,-13): k in [0..n], n in [0..10]]; // A015129 // G. C. Greubel, Dec 01 2021

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

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

flatten([[q_binomial(n,k,-13) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Dec 01 2021

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,...

A015110 Triangle of q-binomial coefficients for q=-3.

Original entry on oeis.org

1, 1, 1, 1, -2, 1, 1, 7, 7, 1, 1, -20, 70, -20, 1, 1, 61, 610, 610, 61, 1, 1, -182, 5551, -15860, 5551, -182, 1, 1, 547, 49777, 433771, 433771, 49777, 547, 1, 1, -1640, 448540, -11662040, 35569222, -11662040, 448540, -1640, 1, 1, 4921, 4035220, 315323620

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), A014983 (k=1), A015251 (k=2), A015268 (k=3), A015288 (k=4), A015306 (k=5), A015324 (k=6), A015340 (k=7), A015357 (k=8), A015375 (k=9), A015388 (k=10), A015407 (k=11), A015424 (k=12),... - M. F. Hasler, Nov 04 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. analog triangles for other q: A015109 (q=-2), 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), A015132 (q=-14), A015133 (q=-15); 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 04 2012

Flatten[Table[QBinomial[n, m, -3], {n, 0, 50}, {m, 0, n}]] (* Vincenzo Librandi, Nov 01 2012 *)

T015110(n, k, q=-3)=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

A015117 Triangle of q-binomial coefficients for q=-7.

Original entry on oeis.org

1, 1, 1, 1, -6, 1, 1, 43, 43, 1, 1, -300, 2150, -300, 1, 1, 2101, 105050, 105050, 2101, 1, 1, -14706, 5149551, -35927100, 5149551, -14706, 1, 1, 102943, 252313293, 12328144851, 12328144851, 252313293, 102943, 1, 1, -720600, 12363454300

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), A014989 (k=1), A015258 (k=2), A015275, A015293, A015312, A015330, A015346, A015363, A015379, A015393 (k=10), A015411, A015430,... - M. F. Hasler, Nov 04 2012

Harvey P. Dale, Table of n, a(n) for n = 0..1000
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), A015118 (q=-8), A015121 (q=-9), A015123 (q=-10), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15);

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 04 2012

Flatten[Table[QBinomial[n,m,-7],{n,0,10},{m,0,n}]] (* Harvey P. Dale, Aug 08 2012 *)

T015117(n, k, q=-7)=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

A015133 Triangle of (Gaussian) q-binomial coefficients for q=-15.

Original entry on oeis.org

1, 1, 1, 1, -14, 1, 1, 211, 211, 1, 1, -3164, 47686, -3164, 1, 1, 47461, 10726186, 10726186, 47461, 1, 1, -711914, 2413439311, -36190151564, 2413439311, -711914, 1, 1, 10678711, 543023133061, 122144174967811, 122144174967811, 543023133061

Offset: 0

Olivier Gérard

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), A015132 (q=-14). - 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

A015133(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

As a triangle, T(n, k) = Product_{i=1..k} ((-15)^(n-i+1)-1)/((-15)^i-1), with 0 <= k <= n = 0,1,2,... - M. F. Hasler, Nov 05 2012

A015112 Triangle of q-binomial coefficients for q=-4.

Original entry on oeis.org

1, 1, 1, 1, -3, 1, 1, 13, 13, 1, 1, -51, 221, -51, 1, 1, 205, 3485, 3485, 205, 1, 1, -819, 55965, -219555, 55965, -819, 1, 1, 3277, 894621, 14107485, 14107485, 894621, 3277, 1, 1, -13107, 14317213, -901984419, 3625623645, -901984419, 14317213, -13107, 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 A000012 (k=0), A014985 (k=1), A015253 (k=2), A015271, A015289, A015308, A015326, A015341, A015359, A015376, A015390 (k=10), A015408, A015425,... - M. F. Hasler, Nov 04 2012

Cf. analog triangles for other q: A015109 (q=-2), A015110 (q=-3), 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), A015132 (q=-14), A015133 (q=-15); 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 04 2012

Flatten[Table[QBinomial[n,m,-4],{n,0,10},{m,0,n}]] (* Harvey P. Dale, Jun 10 2015 *)

T015112(n, k, q=-4)=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

cp's OEIS Frontend

A125813 q-Bell numbers for q=3; eigensequence of A022167, which is the triangle of Gaussian binomial coefficients [n,k] for q=3.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A143777 Eigentriangle of triangle A022167.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A174527 Triangle T(n,m) = 2*A022167(n,m) - binomial(n, m), 0 <= m <= n, read by rows.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A022166 Triangle of Gaussian binomial coefficients (or q-binomial coefficients) [n,k] for q = 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

PARI

Sage

Formula

A015109 Triangle of Gaussian (or q-binomial) coefficients for q = -2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A015129 Triangle of (Gaussian) q-binomial coefficients for q = -13.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A015110 Triangle of q-binomial coefficients for q=-3.