A015397 -id:A015397 - cp's OEIS frontend

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

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

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

A015121 Triangle of q-binomial coefficients for q=-9.

Original entry on oeis.org

1, 1, 1, 1, -8, 1, 1, 73, 73, 1, 1, -656, 5986, -656, 1, 1, 5905, 484210, 484210, 5905, 1, 1, -53144, 39226915, -352504880, 39226915, -53144, 1, 1, 478297, 3177326971, 257015284435, 257015284435, 3177326971, 478297, 1, 1, -4304672, 257363962948

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), A014991 (k=1), A015260 (k=2), A015277 (k=3), A015295 (k=4), A015315 (k=5), A015332 (k=6), A015349 (k=7), A015365 (k=8), A015381 (k=9), A015397 (k=10), A015414 (k=11), A015432 (k=12). - M. F. Hasler, Nov 05 2012

Index entries related to Gaussian binomial coefficients.

Cf. analog 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), A015123 (q=-10), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), 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, 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

Table[QBinomial[n, k, -9], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)

T015121(n, k, q=-9)=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

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

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

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

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

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A015121 Triangle of q-binomial coefficients for q=-9.

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

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

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

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

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