A015379 -id:A015379 - cp's OEIS frontend

A015385 Gaussian binomial coefficient [ n,9 ] for q=-13.

1, -9847035132, 105044442632566365137, -1113436927250681654567602842120, 11807854622717155763702496765310830475383, -125216049699851612689080581288579246248342359563916

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..110
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), A015402 (r=10), A015422 (r=11), A015438 (r=12). - M. F. Hasler, Nov 03 2012

Cf. Gaussian binomial coefficients [n, 9] for q = -2..-13: A015371, A015375, A015376, A015377, A015378, A015379, A015380, A015381, A015382, A015383, A015384. - Vincenzo Librandi, Nov 04 2012

r:=9; q:=-13; [&*[(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, -13],{n, 9, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

A015385(n,r=9,q=-13)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012

a(n) = Product_{i=1..9} ((-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

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

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

A015380 Gaussian binomial coefficient [ n,9 ] for q=-8.

Original entry on oeis.org

1, -119304647, 16266970069380217, -2179059787976052939572615, 292539874786707389459461268654713, -39262839136506665155883080645146897495431, 5269789166381879647128952074697436662720144919161

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..130

Cf. Gaussian binomial coefficients [n, 9] for q = -2..-13: A015371, A015375, A015376, A015377, A015378, A015379, A015381, A015382, A015383, A015384, A015385. - Vincenzo Librandi, Nov 04 2012

r:=9; q:=-8; [&*[(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, -8],{n, 9, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,9,-8) for n in range(9,15)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..9} ((-8)^(n-i+1)-1)/((-8)^i-1). - Vincenzo Librandi, Nov 04 2012

A015381 Gaussian binomial coefficient [ n,9 ] for q=-9.

Original entry on oeis.org

1, -348678440, 136773736379522605, -52916360230556551635386480, 20504007291105533368839949866598015, -7943538006665671364765186721016327317109448, 3077495169782617972230910362141435994555138110002155

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..120

Cf. Gaussian binomial coefficients [n, 9] for q = -2..-13: A015371, A015375, A015376, A015377, A015378, A015379, A015380, A015382, A015383, A015384, A015385. - Vincenzo Librandi, Nov 04 2012

r:=9; q:=-9; [&*[(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, -9],{n, 9, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

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

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

A015382 Gaussian binomial coefficient [ n,9 ] for q=-10.

Original entry on oeis.org

1, -909090909, 918273645463728191, -917356289173636281073462809, 917448033977125729275307703398447191, -917438859588520669588272049420660231320652809, 917439777028298615325746963688293507886210115870347191

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..120

Cf. Gaussian binomial coefficients [n, 9] for q = -2..-13: A015371, A015375, A015376, A015377, A015378, A015379, A015380, A015381, A015383, A015384, A015385. - Vincenzo Librandi, Nov 04 2012

r:=9; q:=-10; [&*[(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, -10],{n, 9, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

a(n) = Product_{i=1..9} ((-10)^(n-i+1)-1)/((-10)^i-1). - Vincenzo Librandi, Nov 04 2012

cp's OEIS Frontend

A015385 Gaussian binomial coefficient [ n,9 ] for q=-13.

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

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

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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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Sage

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

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Sage

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

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Magma