A015402 -id:A015402 - cp's OEIS frontend

A015394 Gaussian binomial coefficient [ n,10 ] for q=-8.

1, 954437177, 1041086085394771065, 1115678612484825190455949945, 1198243328242032079710778546865654393, 1286564714023293732070008866290952083995937401, 1381443612518576172240265744739493702803061753684478585

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..120

Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13: A015386, A015388, A015390, A015391, A015392, A015393, A015397, A015398, A015399, A015401, A015402. - Vincenzo Librandi, Nov 04 2012

r:=10; q:=-8; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 04 2012

Table[QBinomial[n, 10, -8], {n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,10,-8) for n in range(10,16)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..10} ((-8)^(n-i+1)-1)/((-8)^i-1) (by definition). - Vincenzo Librandi, Nov 04 2012

A015397 Gaussian binomial coefficient [ n,10 ] for q=-9.

Original entry on oeis.org

1, 3138105961, 11078672649879436966, 38576026619154398792076180886, 134526791875519431052113309866825757301, 469057975890128020293538941741406421614821552253, 1635507110993502253670495254060345828123783573932476807608

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..110

Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13: A015386, A015388, A015390, A015391, A015392, A015393, A015394, A015398, A015399, A015401, A015402. - Vincenzo Librandi, Nov 04 2012

r:=10; q:=-9; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 04 2012

Table[QBinomial[n, 10, -9], {n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,10,-9) for n in range(10,16)] # Zerinvary Lajos, May 25 2009

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

A015398 Gaussian binomial coefficient [ n,10 ] for q=-10.

Original entry on oeis.org

1, 9090909091, 91827364555463728191, 917356289265463645628926537191, 9174480340688613582018540679613398447191, 91743885968026547299515818524084563811678679347191, 917439777120042501293773510987809326410294679682025870347191

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..110

Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13: A015386, A015388, A015390, A015391, A015392, A015393, A015394, A015397, A015399, A015401, A015402. - Vincenzo Librandi, Nov 04 2012

r:=10; q:=-10; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 04 2012

Table[QBinomial[n, 10, -10], {n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,10,-10) for n in range(10,16)] # Zerinvary Lajos, May 25 2009

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

A015401 Gaussian binomial coefficient [ n,10 ] for q=-12.

Original entry on oeis.org

1, 57154490053, 3563602618051323347605, 220521264778812882986788501660885, 13654753975171772337501943609360145428875733, 845462977543736084817433183822531039414960234418458069

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

Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13: A015386, A015388, A015390, A015391, A015392, A015393, A015394, A015397, A015398, A015399, A015402.

r:=10; q:=-12; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 04 2012

Table[QBinomial[n, 10, -12], {n, 10, 20}] (* Vincenzo Librandi, Nov 05 2012 *)

[gaussian_binomial(n,10,-12) for n in range(10,15)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..10} ((-12)^(n-i+1)-1)/((-12)^i-1) (by definition). - Vincenzo Librandi, Nov 05 2012

A015265 Gaussian binomial coefficient [ n,2 ] for q = -13.

Original entry on oeis.org

1, 157, 26690, 4508570, 761974851, 128773405047, 21762709934980, 3677897920745140, 621564749363392901, 105044442632566365137, 17752510805031727164870, 3000174326048697741925710, 507029461102251552321630151, 85687978926280231101185088427

Offset: 2

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 = 2..200
Index entries for linear recurrences with constant coefficients, signature (157, 2041, -2197).

Cf. Gaussian binomial coefficients [n,2] for q=-2,...,-12: A015249, A015251, A015253, A015255, A015257 A015258, A015259, A015260, A015261, A015262, A015264.

Cf. Gaussian binomial coefficients [n,r] for q=-13: A015286 (r=3), A015303 (r=4), A015321 (r=5), 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

I:=[1,157,26690]; [n le 3 select I[n] else 157*Self(n-1)+2041*Self(n-2)-2197*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 28 2012

Table[QBinomial[n, 2, -13], {n, 2, 20}] (* Vincenzo Librandi, Oct 28 2012 *)

A015265(n,q=-13)=(1-q^n)*(q^(n-1)-1)/2352 \\ M. F. Hasler, Nov 03 2012

[gaussian_binomial(n,2,-13) for n in range(2,14)] # Zerinvary Lajos, May 27 2009

G.f.: x^2/((1-x)*(1+13*x)*(1-169*x)). - Ralf Stephan, Apr 01 2004
a(2) = 1, a(3) = 157, a(4) = 26690, a(n) = 157*a(n-1) + 2041*a(n-2) - 2197*a(n-3). - Vincenzo Librandi, Oct 28 2012
a(n) = (1/2352)*( (1 - (-13)^n)*((-13)^(n-1) - 1) ). - M. F. Hasler, Nov 03 2012

A015286 Gaussian binomial coefficient [ n,3 ] for q = -13.

Original entry on oeis.org

1, -2040, 4508570, -9900819720, 21752862899691, -47790911017216080, 104996653267533662740, -230677643550873536294640, 506798783502833908602716981, -1113436927250681654567602842120

Offset: 3

Olivier Gérard, Dec 11 1999

			A015286(7) = 21752862899691 = A015303(7),
A015286(8) = -47790911017216080 = A015321(8),
A015286(9) = 104996653267533662740 = A015337(9). - _M. F. Hasler_, Nov 03 2012

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 = 3..200
Index entries related to Gaussian binomial coefficients.
Index entries for linear recurrences with constant coefficients, signature (-2040,346970,4481880,-4826809)

Cf. Gaussian binomial coefficients [n,r] for q=-13: A015265 (r=2), A015303 (r=4), A015321 (r=5), 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

Fourth row (r=3) or column (resp. diagonal) in A015129 (read as square array resp. triangle). - M. F. Hasler, Nov 03 2012

r:=3; q:=-13; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 02 2016

QBinomial[Range[3,15],3,-13] (* Harvey P. Dale, Jun 21 2012 *)
Table[QBinomial[n, 3, -13], {n, 3, 20}] (* Vincenzo Librandi, Oct 28 2012 *)

A015286(n,r=3,q=-13)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012

[gaussian_binomial(n,3,-13) for n in range(3,13)] # Zerinvary Lajos, May 27 2009

a(n) = Product_{i=1..3} ((-13)^(n-i+1) - 1)/((-13)^i - 1). - M. F. Hasler, Nov 03 2012

G.f.: x^3 / ( (x-1)*(2197*x+1)*(13*x+1)*(169*x-1) ). - R. J. Mathar, Aug 03 2016

A015303 Gaussian binomial coefficient [ n,4 ] for q = -13.

Original entry on oeis.org

1, 26521, 761974851, 21752862899691, 621305270140974342, 17745052029585350965782, 506816536013640476467362442, 14475186854407942097510802411322

Offset: 4

Olivier Gérard, Dec 11 1999

			To illustrate the relation qC(n,r)=qC(n,n-r), here with r=4, n=r+1...r+3:
A015303(5) = 26521 = A015000(5),
A015303(6) = 761974851 = A015265(6),
A015303(7) = 21752862899691 = A015286(7).

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 = 4..200
Index entries related to Gaussian binomial coefficients.
Index entries for linear recurrences with constant coefficients, signature (26521,58611410,-9905328290,-128011801489,137858491849).

Cf. q-integers and Gaussian binomial coefficients [n,r] for q=-13: A015000, A015265 (r=2), A015286 (r=3), A015321 (r=5), 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

Fifth row (r=4) or column (resp. diagonal) of A015129, read as square (resp. triangular) array.

Table[QBinomial[n, 4, -13], {n, 4, 20}] (* Vincenzo Librandi, Oct 29 2012 *)

A015303(n,r=4,q=-13)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012

[gaussian_binomial(n,4,-13) for n in range(4,12)] # Zerinvary Lajos, May 27 2009

a(n) = product_{i=1..4} ((-13)^(n-i+1)-1)/((-13)^i-1). - M. F. Hasler, Nov 03 2012

G.f.: -x^4 / ( (x-1)*(169*x-1)*(2197*x+1)*(13*x+1)*(28561*x-1) ). - R. J. Mathar, Aug 03 2016

A015337 Gaussian binomial coefficient [ n,6 ] for q = -13.

Original entry on oeis.org

1, 4482037, 21762709934980, 104996653267533662740, 506816536013640476467362442, 2446300028783605805772822454177234, 11807825441932996339362317150047214843540

Offset: 6

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 = 6..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), 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, 6, -13], {n, 6, 10}] (* Vincenzo Librandi, Oct 29 2012 *)

A015337(n,r=6,q=-13)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012

[gaussian_binomial(n,6,-13) for n in range(6,13)] # Zerinvary Lajos, May 27 2009

a(n) = Product_{i=1..6} ((-13)^(n-i+1)-1)/((-13)^i-1). - M. F. Hasler, Nov 03 2012

A015355 Gaussian binomial coefficient [ n,7 ] for q=-13.

Original entry on oeis.org

1, -58266480, 3677897920745140, -230677643550873536294640, 14475186854407942097510802411322, -908294062111964496034866469968025332240

Offset: 7

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 = 7..130
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), A015370 (r=8), A015385 (r=9), A015402 (r=10), A015422 (r=11), A015438 (r=12). - M. F. Hasler, Nov 03 2012

r:=7; q:=-13; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..15]]; // Vincenzo Librandi, Nov 02 2012

Table[QBinomial[n, 7, -13], {n, 7, 16}] (* Vincenzo Librandi, Nov 02 2012 *)

A015355(n,r=7,q=-13)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012

[gaussian_binomial(n,7,-13) for n in range(7,13)] # Zerinvary Lajos, May 27 2009

a(n) = Product_{i=1..7} ((-13)^(n-i+1)-1)/((-13)^i-1). - M. F. Hasler, Nov 03 2012

A015399 Gaussian binomial coefficient [ n,10 ] for q=-11.

Original entry on oeis.org

1, 23775972551, 621826557818118395106, 16116470915170412804822871108406, 418048302457998082359053173653182700919721, 10843028997901257369999365975865569183708813670389271

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

Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13: A015386, A015388, A015390, A015391, A015392, A015393, A015394, A015397, A015398, A015401, A015402.

r:=10; q:=-11; [&*[(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, -11], {n, 10, 20}] (* Vincenzo Librandi, Nov 05 2012 *)

[gaussian_binomial(n,10,-11) for n in range(10,16)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..10} ((-11)^(n-i+1)-1)/((-11)^i-1) (by definition). - Vincenzo Librandi, Nov 05 2012

cp's OEIS Frontend

A015394 Gaussian binomial coefficient [ n,10 ] for q=-8.

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A015397 Gaussian binomial coefficient [ n,10 ] for q=-9.

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A015398 Gaussian binomial coefficient [ n,10 ] for q=-10.

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A015401 Gaussian binomial coefficient [ n,10 ] for q=-12.

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A015265 Gaussian binomial coefficient [ n,2 ] for q = -13.

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A015286 Gaussian binomial coefficient [ n,3 ] for q = -13.

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A015303 Gaussian binomial coefficient [ n,4 ] for q = -13.

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