A015402 -id:A015402 - 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

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

Original entry on oeis.org

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

A015388 Gaussian binomial coefficient [ n,10 ] for q=-3.

Original entry on oeis.org

1, 44287, 2941985410, 167517069529030, 10015359787639069513, 588973263031690760850991, 34826053765400471578213696840, 2055503791013087031667210071738520, 121393945396362834176064326157233601646

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..200

Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13: A015386, A015390, A015391, A015392, A015393, A015394, A015397, A015398, A015399, A015401, A015402.

r:=10; q:=-3; [&*[(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, -3], {n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,10,-3) for n in range(10,18)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..10} ((-3)^(n-i+1)-1)/((-3)^i-1) (by definition). - Vincenzo Librandi, 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,...

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

A015390 Gaussian binomial coefficient [ n,10 ] for q=-4.

Original entry on oeis.org

1, 838861, 938250090141, 968690748238618461, 1019729183363623510391901, 1068220365220113899181567068253, 1120383768613759382944995805859747933, 1174735830441360695151745376566623493806173, 1231818594183047090443637654682442929123639613533

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..150

Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13: A015386, A015388, A015391, A015392, A015393, A015394, A015397, A015398, A015399, A015401, A015402. - Vincenzo Librandi, Nov 04 2012

r:=10; q:=-4; [&*[(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, -4], {n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,10,-4) for n in range(10,17)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..10} ((-4)^(n-i+1)-1)/((-4)^i-1) (by definition). - Vincenzo Librandi, Nov 04 2012

G.f.: x^10 / ((x-1) * (4*x+1) * (16*x-1) * (64*x+1) * (256*x-1) * (1024*x+1) * (4096*x-1) * (16384*x+1) * (65536*x-1) * (262144*x+1) * (1048576*x-1)). - Colin Barker, Jan 13 2014

A015391 Gaussian binomial coefficient [ n,10 ] for q=-5.

Original entry on oeis.org

1, 8138021, 82784230211046, 802023560334345174046, 7844813030956382105126218421, 76584995059524711257676812461230921, 747948211058777330441088769852487456090296, 7304088256300765454892487244083619479306573590296

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..150

Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13: A015386, A015388, A015390, A015392, A015393, A015394, A015397, A015398, A015399, A015401, A015402.

r:=10; q:=-5; [&*[(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, -5], {n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,10,-5) for n in range(10,17)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..10} ((-5)^(n-i+1)-1)/((-5)^i-1) (by definition). - Vincenzo Librandi, Nov 04 2012

A015386 Gaussian binomial coefficient [ n,10 ] for q=-2.

Original entry on oeis.org

1, 683, 932295, 848699215, 926949282623, 920460637644639, 957498220445101855, 972884994173649887135, 1000137219716325891620511, 1022146087305755916943130783, 1047699739488399814866709052575, 1072321450350081081965428740719775

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..200
Index entries for linear recurrences with constant coefficients, signature (683,465806,-106203768, -14443712448,903388560384,28908433932288,-473291569496064, -3563607111499776,16004972290244608,24030926136672256,-36028797018963968).

Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13: A015388, A015390, A015391, A015392, A015393, A015394, A015397, A015398, A015399, A015401, A015402.

r:=10; q:=-2; [&*[(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, -2],{n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,10,-2) for n in range(10,21)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..10} ((-2)^(n-i+1)-1)/((-2)^i-1) (by definition). - Vincenzo Librandi, Nov 04 2012

G.f.: x^10 / ( (x-1)*(512*x+1)*(64*x-1)*(128*x+1)*(1024*x-1)*(2*x+1)*(8*x+1)*(32*x+1)*(16*x-1)*(4*x-1)*(256*x-1) ). - R. J. Mathar, Sep 22 2016

A015392 Gaussian binomial coefficient [ n,10 ] for q=-6.

Original entry on oeis.org

1, 51828151, 3223388672928931, 194007802557550502202331, 11739968552378570066280405695371, 709779726467093092873777345973423761771, 42918585756017923252384776090351752769462732331

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..140

Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13: A015386, A015388, A015390, A015391, A015393, A015394, A015397, A015398, A015399, A015401, A015402.

r:=10; q:=-6; [&*[(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, -6], {n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,10,-6) for n in range(10,16)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..10} ((-6)^(n-i+1)-1)/((-6)^i-1) (by definition). - Vincenzo Librandi, Nov 04 2012

A015393 Gaussian binomial coefficient [ n,10 ] for q=-7.

Original entry on oeis.org

1, 247165843, 71272779562356450, 20074270583791406305395150, 5672847283550509352791825564114953, 1602343611088456383646516751967506297398179, 452626257785468649545785666454333613632908777305800

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..130

Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13: A015386, A015388, A015390, A015391, A015392, A015394, A015397, A015398, A015399, A015401, A015402. - Vincenzo Librandi, Nov 04 2012

r:=10; q:=-7; [&*[(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, -7], {n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,10,-7) for n in range(10,16)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..10} ((-7)^(n-i+1)-1)/((-7)^i-1) (by definition). - Vincenzo Librandi, Nov 04 2012

cp's OEIS Frontend

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A015388 Gaussian binomial coefficient [ n,10 ] for q=-3.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A015390 Gaussian binomial coefficient [ n,10 ] for q=-4.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A015391 Gaussian binomial coefficient [ n,10 ] for q=-5.

Views

Author

Keywords