A015386 -id:A015386 - cp's OEIS frontend

A015109 Triangle of Gaussian (or q-binomial) coefficients for q = -2.

1, 1, 1, 1, -1, 1, 1, 3, 3, 1, 1, -5, 15, -5, 1, 1, 11, 55, 55, 11, 1, 1, -21, 231, -385, 231, -21, 1, 1, 43, 903, 3311, 3311, 903, 43, 1, 1, -85, 3655, -25585, 56287, -25585, 3655, -85, 1, 1, 171, 14535, 208335, 875007, 875007, 208335, 14535, 171, 1, 1, -341, 58311

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), A077925 (k=1), A015249 (k=2), A015266 (k=3), A015287 (k=4), A015305 (k=5), A015323 (k=6), A015338 (k=7), A015356 (k=8), A015371 (k=9), A015386 (k=10), A015405 (k=11), A015423 (k=12), ... - M. F. Hasler, Nov 04 2012
The elements of the inverse matrix are apparently T^(-1)(n,k) = (-1)^n*A157785(n,k). - R. J. Mathar, Mar 12 2013
Fu et al. give two combinatorial interpretations of the (unsigned) q-binomial coefficients when q is a negative integer. - Peter Bala, Nov 02 2017

			From _Roger L. Bagula_, Feb 10 2009: (Start)
  1;
  1,   1;
  1,  -1,     1;
  1,   3,     3,      1;
  1,  -5,    15,     -5,      1;
  1,  11,    55,     55,     11,      1;
  1, -21,   231,   -385,    231,    -21,      1;
  1,  43,   903,   3311,   3311,    903,     43,     1;
  1, -85,  3655, -25585,  56287, -25585,   3655,   -85,   1;
  1, 171, 14535, 208335, 875007, 875007, 208335, 14535, 171, 1;  (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
J. A. de Azcarraga and J. A. Macfarlane, Group Theoretical Foundations of Fractional Supersymmetry, arxiv:hep-th/9506177 (1995).
S. Fu, V. Reiner, D. Stanton and N. Thiem, The negative q-binomial, arXiv:1108.4702 [math.CO], 2011.
R. Parthasarathy, q-Fermionic Numbers and Their Roles in Some Physical Problems, arxiv:quant-ph/0403216, 2004.

Cf. A015152 (row sums).
Cf. 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).
Analogous triangles for other q: 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), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15).

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,-2): k in [0..n], n in [0..10]]; // A015109 // G. C. Greubel, Nov 30 2021

A015109 := proc(n, k)
   mul( ((-2)^(1+n-i)-1)/((-2)^i-1) ,i=1..k) ;
end proc: # R. J. Mathar, Mar 12 2013

T[n_, k_, q_]:= Product[(1 - q^(n-j+1))/(1 - q^j), {j, k}];
Table[T[n,k,-2], {n,0,10}, {k,0,n}]//Flatten (* Roger L. Bagula, Feb 10 2009 *)(* modified by G. C. Greubel, Nov 30 2021 *)
Table[QBinomial[n, k, -2], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, Apr 09 2016 *)

T015109(n, k, q=-2)=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

flatten([[q_binomial(n,k,-2) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Nov 30 2021

T(n, k) = q-binomial(n, k, -2).

T(n, k, q) = Product_{j=1..k} ( (1 - q^(n-j+1))/(1 - q^j) ), for q = -2. - Roger L. Bagula, Feb 10 2009

Edited by M. F. Hasler, Nov 04 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

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

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

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

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

cp's OEIS Frontend

A015109 Triangle of Gaussian (or q-binomial) coefficients for q = -2.

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

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

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