A015251 -id:A015251 - 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

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

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

A239577 Expansion of 1/((x-1)(3x-1)(3x^2+1)).

Original entry on oeis.org

1, 4, 10, 28, 91, 280, 820, 2440, 7381, 22204, 66430, 199108, 597871, 1794160, 5380840, 16140880, 48427561, 145287604, 435848050, 1307529388, 3922632451, 11767941640, 35303692060, 105910943320, 317733228541, 953200084204, 2859599056870, 8578795974868

Offset: 0

Philippe Deléham, Mar 21 2014

			Ternary................Decimal
1............................1
11...........................4
101.........................10
1001........................28
10101.......................91
101101.....................280
1010101....................820
10100101..................2440
101010101.................7381
1010110101...............22204
10101010101..............66430
101010010101............199108, etc.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, -6, 12, -9).

Cf. A002452, A015251, A147759, A154957.

Table[(-1 + 3^(2 + n) + (-1 + (-1)^n) (-3)^((1 + n)/2))/8, {n, 0, 30}] (* Bruno Berselli, Mar 24 2014 *)
CoefficientList[Series[1/((x - 1) (3 x - 1) (3 x^2 + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)
LinearRecurrence[{4,-6,12,-9},{1,4,10,28},30] (* Harvey P. Dale, Oct 04 2024 *)

G.f.: 1/((x-1)*(3*x-1)*(3*x^2+1)).
a(n) = Sum{k=0..n} A154957(n,k)*3^k.
a(n) = 4*a(n-1) - 6*a(n-2) + 12*a(n-3) - 9*a(n-4) for n > 3, a(0)=1, a(1)=4, a(2)=10, a(3)=16.
a(2*n) = A002452(n+1); a(2*n+1) = 4*A015251(n+2).
a(n) = ( -1 + 3^(2+n) + (-1+(-1)^n)*(-3)^((1+n)/2) )/8. [Bruno Berselli, Mar 24 2014]

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

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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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A239577 Expansion of 1/((x-1)(3x-1)(3x^2+1)).