A015385 -id:A015385 - cp's OEIS frontend

A015370 Gaussian binomial coefficient [ n,8 ] for q=-13.

1, 757464241, 621564749363392901, 506798783502833908602716981, 413425812255544017749839936272484623, 337243227617163445881817693983677965955870943, 275099718210633054941121644140453635236773122223471523

Offset: 8

Olivier Gérard

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 = 8..100
Index entries related to Gaussian binomial coefficients.

Cf. Gaussian binomial coefficients [n,8] for q=-2..-12: A015356, A015357, A015359, A015360, A015361, A015363, A015364, A015365, A015367, A015368, A015369. - M. F. Hasler, Nov 03 2012

Cf. Gaussian binomial coefficients [n,r] for q=-13: A015265 (r=2), A015286 (r=3), A015303 (r=4), A015321 (r=5), A015337 (r=6), A015355 (r=7), A015385 (r=9), A015402 (r=10), A015422 (r=11), A015438 (r=12). - M. F. Hasler, Nov 03 2012

r:=8; q:=-13; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..18]]; // Vincenzo Librandi, Nov 03 2012

Table[QBinomial[n, 8, -13], {n, 8, 14}] (* Vincenzo Librandi, Nov 03 2012 *)

A015370(n,r=8,q=-13)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012

[gaussian_binomial(n,8,-13) for n in range(8,14)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..8} ((-13)^(n-i+1)-1)/((-13)^i-1). - M. F. Hasler, Nov 03 2012

A015402 Gaussian binomial coefficient [ n,10 ] for q=-13.

Original entry on oeis.org

1, 128011456717, 17752510805031727164870, 2446220929187500105890055171302510, 337244135881870906696294510219932684378716373, 46491842741544248966048667175076748587505712393943779761

Offset: 10

Olivier Gérard

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 = 10..100
Index entries related to Gaussian binomial coefficients.

Cf. Gaussian binomial coefficients [n,r] for q=-13: A015265 (r=2), A015286 (r=3), A015303 (r=4), A015321 (r=5), A015337 (r=6), A015355 (r=7), A015370 (r=8), A015385 (r=9), A015422 (r=11), A015438 (r=12). - M. F. Hasler, Nov 03 2012

Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13: A015386, A015388, A015390, A015391, A015392, A015393, A015394, A015397, A015398, A015399, A015401. - Vincenzo Librandi, Nov 05 2012

r:=10; q:=-13; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 05 2012

Table[QBinomial[n, 10, -13], {n, 10, 20}] (* Vincenzo Librandi, Nov 05 2012 *)

A015402(n,r=10,q=-13)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012

[gaussian_binomial(n,10,-13) for n in range(10,15)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..10} ((-13)^(n-i+1)-1)/((-13)^i-1). - M. F. Hasler, Nov 03 2012

A015375 Gaussian binomial coefficient [ n,9 ] for q=-3.

Original entry on oeis.org

1, -14762, 326882347, -6204226946060, 123644349019377043, -2423717068608654822146, 47771556642163840723529281, -939857780045414554730512966640, 18502040831058043147238631145734166, -364157167636884405223950738210339855212

Offset: 9

Olivier Gérard

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 = 9..200

Cf. Gaussian binomial coefficients [n,9] for q=-2..-13: A015371, A015376, A015377, A015378, A015379, A015380, A015381, A015382, A015383, A015384, A015385. - Vincenzo Librandi, Nov 04 2012

Gaussian binomial coefficient [n, k]_q for q = -3: A015251 (k = 2), A015268 (k = 3), A015288 (k = 4), A015306 (k = 5), A015324 (k = 6), A015340 (k = 7), A015357 (k = 8), this sequence (k = 9), A015388 (k = 10).

r:=9; q:=-3; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 04 2012

Table[QBinomial[n, 9, -3],{n, 9, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,9,-3) for n in range(9,18)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..9} ((-3)^(n-i+1)-1)/((-3)^i-1) (by definition). - Vincenzo Librandi, Nov 04 2012
G.f.: -x^9 / ( (x-1)*(27*x+1)*(81*x-1)*(729*x-1)*(9*x-1)*(2187*x+1)*(3*x+1)*(19683*x+1)*(6561*x-1)*(243*x+1) ). - R. J. Mathar, Sep 02 2016
G.f. with offset 0: exp(Sum_{n >= 1} A015518(10*n)/A015518(n) * (-x)^n/n) = 1 - 14762*x + 326882347*x^2 + .... - Peter Bala, Jun 29 2025

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,...

A015321 Gaussian binomial coefficient [ n,5 ] for q = -13.

Original entry on oeis.org

1, -344772, 128773405047, -47790911017216080, 17745052029585350965782, -6588595858168804130787130344, 2446300028783605805772822454177234, -908294062111964496034866469968025332240

Offset: 5

Olivier Gérard, Dec 11 1999

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 = 5..180
Index entries related to Gaussian binomial coefficients.

Cf. Gaussian binomial coefficients [n,r] for q=-13: A015265 (r=2), A015286 (r=3), A015303 (r=4), A015337 (r=6), A015355 (r=7), A015370 (r=8), A015385 (r=9), A015402 (r=10), A015422 (r=11), A015438 (r=12). - M. F. Hasler, Nov 03 2012

Table[QBinomial[n, 5, -13], {n, 5, 20}] (* Vincenzo Librandi, Oct 29 2012 *)

A015321(n,r=5,q=-13)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012

[gaussian_binomial(n,5,-13) for n in range(5,13)] # Zerinvary Lajos, May 27 2009

a(n) = Product_{i=1..5} ((-13)^(n-i+1)-1)/((-13)^i-1). - M. F. Hasler, Nov 03 2012

A015371 Gaussian binomial coefficient [ n,9 ] for q=-2.

Original entry on oeis.org

1, -341, 232903, -105970865, 57881286463, -28735427761313, 14946527496991519, -7593183562134412385, 3902985682508407194271, -1994425683761796076272481, 1022146087305755916943130783, -523082886040328458081329117025

Offset: 9

Olivier Gérard

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 = 9..200

Diagonal k=9 of the triangular array A015109. See there for further references and programs. - M. F. Hasler, Nov 04 2012

Cf. Gaussian binomial coefficients [n,9] for q=-2..-13: A015375, A015376, A015377, A015378, A015379, A015380, A015381, A015382, A015383, A015384, A015385. - Vincenzo Librandi, Nov 04 2012

r:=9; q:=-2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 04 2012

Table[QBinomial[n, 9, -2],{n, 9, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,9,-2) for n in range(9,21)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..9} ((-2)^(n-i+1)-1)/((-2)^i-1). - Vincenzo Librandi, Nov 04 2012

G.f.: -x^9 / ( (x-1)*(512*x+1)*(64*x-1)*(128*x+1)*(2*x+1)*(8*x+1)*(32*x+1)*(16*x-1)*(4*x-1)*(256*x-1) ). - R. J. Mathar, Sep 02 2016

A015376 Gaussian binomial coefficient [ n,9 ] for q=-4.

Original entry on oeis.org

1, -209715, 58640578205, -15135778281070755, 3983313338565919030365, -1043182954580986851130914723, 273530932713230996784935699290205, -71700116580663579186545558567554787235, 18796042166858164201094703719132482337953885

Offset: 9

Olivier Gérard

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 = 9..190

Cf. Gaussian binomial coefficients [n,9] for q=-2..-13: A015371, A015375, A015377,A015378, A015379, A015380, A015381, A015382, A015383, A015384, A015385. - Vincenzo Librandi, Nov 04 2012

r:=9; q:=-4; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 04 2012

Table[QBinomial[n, 9, -4],{n, 9, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,9,-4) for n in range(9,17)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..9} ((-4)^(n-i+1)-1)/((-4)^i-1) (by definition). - Vincenzo Librandi, Nov 04 2012

G.f.: -x^9 / ( (x-1)*(16384*x+1)*(4096*x-1)*(256*x-1)*(65536*x-1)*(64*x+1)*(262144*x+1)*(4*x+1)*(16*x-1)*(1024*x+1) ). - R. J. Mathar, Sep 02 2016

A015377 Gaussian binomial coefficient [ n,9 ] for q=-5.

Original entry on oeis.org

1, -1627604, 3311368882921, -6416187820400919704, 12551699566292514833249671, -24507195908707737696414306347204, 47868680606322065338648160779243199671, -93492320106912696270274007078334075223284704

Offset: 9

Olivier Gérard

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 = 9..170

Cf. Gaussian binomial coefficients [n,9] for q=-2..-13: A015371, A015375, A015376, A015378, A015379, A015380, A015381, A015382, A015383, A015384, A015385. - Vincenzo Librandi, Nov 04 2012

r:=9; q:=-5; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 04 2012

Table[QBinomial[n, 9, -5], {n, 9, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,9,-5) for n in range(9,16)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..9} ((-5)^(n-i+1)-1)/((-5)^i-1) (by definition). - Vincenzo Librandi, Nov 04 2012

A015378 Gaussian binomial coefficient [ n,9 ] for q=-6.

Original entry on oeis.org

1, -8638025, 89538572808355, -898184256176675135525, 9058617560471271225871839115, -91278255494743382265330154281509525, 919894226814090294609303909820267635374635, -9270381253910297854571803793049953719997957501525

Offset: 9

Olivier Gérard

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 = 9..150

Cf. Gaussian binomial coefficients [n,9] for q = -2..-13: A015371, A015375, A015376, A015377, A015379, A015380, A015381, A015382, A015383, A015384, A015385. - Vincenzo Librandi, Nov 04 2012

r:=9; q:=-6; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 04 2012

QBinomial[Range[9,20],9,-6] (* Harvey P. Dale, Aug 16 2012 *)
Table[QBinomial[n, 9, -6],{n, 9, 18}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,9,-6) for n in range(9,16)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..9} ((-6)^(n-i+1)-1)/((-6)^i-1). - Vincenzo Librandi, Nov 04 2012

A015379 Gaussian binomial coefficient [ n,9 ] for q=-7.

Original entry on oeis.org

1, -35309406, 1454546516636543, -58525570007342935110900, 2362701900656492615160524472603, -95337871447349860183019420430515900118, 3847259697771549596318959641032366290112134229

Offset: 9

Olivier Gérard

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 = 9..140

Cf. Gaussian binomial coefficients [n, 9] for q = -2..-13: A015371, A015375, A015376, A015377, A015378, A015380, A015381, A015382, A015383, A015384, A015385. - Vincenzo Librandi, Nov 04 2012

r:=9; q:=-7; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 04 2012

Table[QBinomial[n, 9, -7],{n, 9, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,9,-7) for n in range(9,15)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..9} ((-7)^(n-i+1)-1)/((-7)^i-1). - Vincenzo Librandi, Nov 04 2012

cp's OEIS Frontend

A015370 Gaussian binomial coefficient [ n,8 ] for q=-13.

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A015402 Gaussian binomial coefficient [ n,10 ] for q=-13.

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A015375 Gaussian binomial coefficient [ n,9 ] for q=-3.

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

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A015321 Gaussian binomial coefficient [ n,5 ] for q = -13.

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A015371 Gaussian binomial coefficient [ n,9 ] for q=-2.

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A015376 Gaussian binomial coefficient [ n,9 ] for q=-4.

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