A015288 -id:A015288 - cp's OEIS frontend

A015357 Gaussian binomial coefficient [ n,8 ] for q=-3.

1, 4921, 36321901, 229798289941, 1526550040078063, 9974653139743515223, 65533580739687859229563, 429769342296322230713871283, 2820146424148466477944423359046, 18502040831058043147238631145734166

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..200
Index entries for linear recurrences with constant coefficients, signature (4921,12105660,-8513737740,-2091825362718,169437854380158,4524549298283340,-42209826451809660,-112576695670863081,150094635296999121).

Cf. Gaussian binomial coefficients [n,8] for q=-2..-13: A015356, A015359, A015360, A015361, A015363, A015364, A015365, A015367, A015368, A015369, A015370. - M. F. Hasler, Nov 03 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), this sequence (k = 8), A015375 (k = 9), A015388 (k = 10).

r:=8; q:=-3; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 02 2012

Table[QBinomial[n, 8, -3], {n, 8, 20}] (* Vincenzo Librandi, Nov 02 2012 *)

A015357(n, r=8, q=-3)=prod(i=1, r, (1-q^(n-i+1))/(1-q^i)) \\ M. F. Hasler, Nov 03 2012

[gaussian_binomial(n,8,-3) for n in range(8,18)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..8} ((-3)^(n-i+1)-1)/((-3)^i-1). - M. F. Hasler, Nov 03 2012
G.f.: -x^8 / ( (x-1)*(27*x+1)*(81*x-1)*(729*x-1)*(9*x-1)*(2187*x+1)*(3*x+1)*(6561*x-1)*(243*x+1) ). - R. J. Mathar, Sep 02 2016
G.f. with offset 0: exp(Sum_{n >= 1} A015518(9*n)/A015518(n) * x^n/n) = 1 + 4921*x + 36321901*x^2 + .... - Peter Bala, Jun 29 2025

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

A015110 Triangle of q-binomial coefficients for q=-3.

Original entry on oeis.org

1, 1, 1, 1, -2, 1, 1, 7, 7, 1, 1, -20, 70, -20, 1, 1, 61, 610, 610, 61, 1, 1, -182, 5551, -15860, 5551, -182, 1, 1, 547, 49777, 433771, 433771, 49777, 547, 1, 1, -1640, 448540, -11662040, 35569222, -11662040, 448540, -1640, 1, 1, 4921, 4035220, 315323620

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), A014983 (k=1), A015251 (k=2), A015268 (k=3), A015288 (k=4), A015306 (k=5), A015324 (k=6), A015340 (k=7), A015357 (k=8), A015375 (k=9), A015388 (k=10), A015407 (k=11), A015424 (k=12),... - M. F. Hasler, Nov 04 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. analog triangles for other q: A015109 (q=-2), 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); 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, -3], {n, 0, 50}, {m, 0, n}]] (* Vincenzo Librandi, Nov 01 2012 *)

T015110(n, k, q=-3)=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

A015251 Gaussian binomial coefficient [ n,2 ] for q = -3.

Original entry on oeis.org

1, 7, 70, 610, 5551, 49777, 448540, 4035220, 36321901, 326882347, 2941985410, 26477735830, 238300021051, 2144698993717, 19302294530680, 173720640014440, 1563485792415001, 14071372034879887

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.

G. C. Greubel, Table of n, a(n) for n = 2..500
Index entries for linear recurrences with constant coefficients, signature (7,21,-27).

Gaussian binomial coefficient [n, k]_q for q = -3: A015268 (k = 3), A015288 (k = 4), A015306 (k = 5), A015324 (k = 6), A015340 (k = 7), A015357 (k = 8), A015375 (k = 9), A015388 (k = 10).

Cf. A015518.

Table[QBinomial[n, 2, -3], {n, 2, 25}] (* G. C. Greubel, Jul 30 2016 *)

a(n)=([0,1,0; 0,0,1; -27,21,7]^(n-2)*[1;7;70])[1,1] \\ Charles R Greathouse IV, Jul 30 2016

[gaussian_binomial(n,2,-3) for n in range(2,18)] # Zerinvary Lajos, May 28 2009

G.f.: x^2/[(1-x)(1+3x)(1-9x)].
a(n) = 10*a(n-1) - 9*a(n-2) + (-1)^n *3^(n-2), n >= 4. - Vincenzo Librandi, Mar 20 2011
a(n) = 7*a(n-1) + 21*a(n-2) - 27*a(n-3), n >= 3. - Vincenzo Librandi, Mar 20 2011
a(n) = (1/96)*(2*(-1)^n*3^n - 3 + 9^n). - R. J. Mathar, Mar 21 2011
G.f. with offset 0: exp(Sum_{n >= 1} A015518(3*n)/A015518(n) * x^n/n) = 1 + 7*x + 70*x^2 + .... - Peter Bala, Jun 29 2025

A015306 Gaussian binomial coefficient [ n,5 ] for q = -3.

Original entry on oeis.org

1, -182, 49777, -11662040, 2869444942, -694405675964, 168973319623174, -41041673208656120, 9974653139743515223, -2423717068608654822146, 588973263031690760850991, -143119691677080990521708240

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..200
Index entries for linear recurrences with constant coefficients, signature (-182,16653,428220,-4046679,-10746918,14348907)

Gaussian binomial coefficient [n, k]_q for q = -3: A015251 (k = 2), A015268 (k = 3), A015288 (k = 4), this sequence (k = 5), A015324 (k = 6), A015340 (k = 7), A015357 (k = 8), A015375 (k = 9), A015388 (k = 10).

Gaussian binomial coefficients [n,5]: A015305 (q=-2), this sequence (q=-3), A015308 (q=-4), A015309 (q=-5), A015310 (q=-6), A015312 (q=-7), A015313 (q=-8), A015315 (q=-9), A015316 (q=-10), A015317 (q=-11), A015319 (q=-12), A015321 (q=-13).

List([5..25], n-> (1 -61*(-3)^(n-4) +610*(-3)^(2*n-7) - 610*(-3)^(3*n-9) +61*(-3)^(4*n-10) -(-3)^(5*n-10))/17489920); # G. C. Greubel, Sep 21 2019

[(1 - 61*(-3)^(n-4) + 610*(-3)^(2*n-7) - 610*(-3)^(3*n-9) + 61*(-3)^(4*n-10) - (-3)^(5*n-10))/17489920: n in [5..25]]; // G. C. Greubel, Sep 21 2019

seq((1 - 61*(-3)^(n-4) + 610*(-3)^(2*n-7) - 610*(-3)^(3*n-9) + 61*(-3)^(4*n-10) - (-3)^(5*n-10))/17489920, n=5..25); # G. C. Greubel, Sep 21 2019

Table[QBinomial[n, 5, -3], {n, 5, 20}] (* Vincenzo Librandi, Oct 29 2012 *)

a(n) = (1 - 61*(-3)^(n-4) + 610*(-3)^(2*n-7) - 610*(-3)^(3*n-9) + 61*(-3)^(4*n-10) - (-3)^(5*n-10))/17489920 \\ G. C. Greubel, Sep 21 2019

[gaussian_binomial(n,5,-3) for n in range(5,17)] # Zerinvary Lajos, May 27 2009

G.f.: x^5/((1-x)*(1+3*x)*(1-9*x)*(1+27*x)*(1-81*x)*(1+243*x)). - R. J. Mathar, Aug 03 2016
From G. C. Greubel, Sep 21 2019: (Start)
a(n) = (1 - 61*(-3)^(n-4) + 610*(-3)^(2*n-7) - 610*(-3)^(3*n-9) + 61*(-3)^(4*n-10) - (-3)^(5*n-10))/17489920.
E.g.f.: exp(-243*x)*(-1 +1830*exp(216*x) -44469*exp(240*x) +59049*exp(244 *x) -16470*exp(252*x) +61*exp(324*x))/1032762286080. (End)
G.f. with offset 0: exp(Sum_{n >= 1} A015518(6*n)/A015518(n) * (-x)^n/n) = 1 - 182*x + 49777*x^2 - .... - Peter Bala, Jun 29 2025

A015268 Gaussian binomial coefficient [ n,3 ] for q = -3.

Original entry on oeis.org

1, -20, 610, -15860, 433771, -11662040, 315323620, -8509702520, 229798289941, -6204226946060, 167517069529030, -4522934399547980, 122119467087816511, -3297223466672052080, 89025052902439936840, -2403676254645238280240

Offset: 3

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 = 3..200
Index entries for linear recurrences with constant coefficients, signature (-20,210,540,-729).

Gaussian binomial coefficient [n, k]_q for q = -3: A015251 (k = 2), this sequence (k = 3), A015288 (k = 4), A015306 (k = 5), A015324 (k = 6), A015340 (k = 7), A015357 (k = 8), A015375 (k = 9), A015388 (k = 10).

Cf. A015518.

[(-1+7*3^(2*n-3)+(-1)^n*3^(n-2)*(7-3^(2*n-1)))/896: n in [3..18]]; // Bruno Berselli, Oct 29 2012

Table[QBinomial[n, 3, -3], {n, 3, 20}] (* Vincenzo Librandi, Oct 28 2012 *)

makelist(coeff(taylor(1/((1-x)*(1+3*x)*(1-9*x)*(1+27*x)), x, 0, n), x, n), n, 0, 15); /* Bruno Berselli, Oct 29 2012 */

[gaussian_binomial(n,3,-3) for n in range(3,19)] # Zerinvary Lajos, May 27 2009

G.f.: x^3/((1-x)*(1+3*x)*(1-9*x)*(1+27*x)). - Bruno Berselli, Oct 29 2012
a(n) = (-1 + 7*3^(2n-3) + (-1)^n*3^(n-2)*(7-3^(2n-1)))/896. - Bruno Berselli, Oct 29 2012
G.f.: x^3 * exp(Sum_{n >= 1} A015518(4*n)/A015518(n) * (-x)^n/n) = x^3 * (1 - 20*x + 610*x^2 - ...). - Peter Bala, Jun 29 2025

A015324 Gaussian binomial coefficient [ n,6 ] for q = -3.

Original entry on oeis.org

1, 547, 448540, 315323620, 232740363922, 168973319623174, 123350523324917020, 89881489830655851460, 65533580739687859229563, 47771556642163840723529281, 34826053765400471578213696840

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..200
Index entries for linear recurrences with constant coefficients, signature (547,149331,-11711817,-316219059,2939282073,7848852129,-10460353203).

Gaussian binomial coefficient [n, k]_q for q = -3: A015251 (k = 2), A015268 (k = 3), A015288 (k = 4), A015306 (k = 5), this sequence (k = 6), A015340 (k = 7), A015357 (k = 8), A015375 (k = 9), A015388 (k = 10).

Table[QBinomial[n, 6, -3], {n, 6, 20}] (* Vincenzo Librandi, Oct 29 2012 *)

[gaussian_binomial(n,6,-3) for n in range(6,17)] # Zerinvary Lajos, May 27 2009

G.f.: x^6 / ( (x-1)*(27*x+1)*(81*x-1)*(729*x-1)*(9*x-1)*(3*x+1)*(243*x+1) ). - R. J. Mathar, Aug 04 2016

G.f. with offset 0: exp(Sum_{n >= 1} A015518(7*n)/A015518(n) * (-x)^n/n) = 1 + 547*x + 448540*x^2 + .... - Peter Bala, Jun 29 2025

A015340 Gaussian binomial coefficient [ n,7 ] for q = -3.

Original entry on oeis.org

1, -1640, 4035220, -8509702520, 18843459775162, -41041673208656120, 89881489830655851460, -196480936769813691291560, 429769342296322230713871283, -939857780045414554730512966640

Offset: 7

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 = 7..200
Index entries for linear recurrences with constant coefficients, signature (-1640,1345620,314875080,-25929962838,-688631799960,6436058745780,17154979252920,-22876792454961).

Gaussian binomial coefficient [n, k]_q for q = -3: A015251 (k = 2), A015268 (k = 3), A015288 (k = 4), A015306 (k = 5), A015324 (k = 6), this sequence (k = 7), A015357 (k = 8), A015375 (k = 9), A015388 (k = 10).

Table[QBinomial[n, 7, -3], {n, 7, 20}] (* Vincenzo Librandi, Oct 29 2012 *)

[gaussian_binomial(n,7,-3) for n in range(7,17)] # Zerinvary Lajos, May 27 2009

G.f.: x^7 / ( (x-1)*(27*x+1)*(81*x-1)*(729*x-1)*(9*x-1)*(2187*x+1)*(3*x+1)*(243*x+1) ). - R. J. Mathar, Sep 02 2016

G.f. with offset 0: exp(Sum_{n >= 1} A015518(8*n)/A015518(n) * (-x)^n/n) = 1 - 1640*x + 4035220*x^2 - .... - Peter Bala, Jun 29 2025

cp's OEIS Frontend

A015357 Gaussian binomial coefficient [ n,8 ] for q=-3.

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

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A015110 Triangle of q-binomial coefficients for q=-3.

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A015251 Gaussian binomial coefficient [ n,2 ] for q = -3.

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A015306 Gaussian binomial coefficient [ n,5 ] for q = -3.

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A015268 Gaussian binomial coefficient [ n,3 ] for q = -3.

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A015324 Gaussian binomial coefficient [ n,6 ] for q = -3.

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