cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

A015382 Gaussian binomial coefficient [ n,9 ] for q=-10.

Original entry on oeis.org

1, -909090909, 918273645463728191, -917356289173636281073462809, 917448033977125729275307703398447191, -917438859588520669588272049420660231320652809, 917439777028298615325746963688293507886210115870347191
Offset: 9

Views

Author

Olivier Gérard

Keywords

References

  • 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.

Links

Crossrefs

Cf. Gaussian binomial coefficients [n, 9] for q = -2..-13: A015371, A015375, A015376, A015377, A015378, A015379, A015380, A015381, A015383, A015384, A015385. - Vincenzo Librandi, Nov 04 2012

Programs

  • Magma
    r:=9; q:=-10; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 04 2012
  • Mathematica
    Table[QBinomial[n, 9, -10],{n, 9, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

Formula

a(n) = Product_{i=1..9} ((-10)^(n-i+1)-1)/((-10)^i-1). - Vincenzo Librandi, Nov 04 2012

A015383 Gaussian binomial coefficient [ n,9 ] for q=-11.

Original entry on oeis.org

1, -2161452050, 5139062461110267955, -12108543136400139930131294300, 28553261556033167915025118560778623715, -67326679110860591163925513616845073983121067050, 158752877164012182076561255078472431325233637546101158985
Offset: 9

Views

Author

Olivier Gérard

Keywords

References

  • 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.

Links

Crossrefs

Cf. Gaussian binomial coefficients [n, 9] for q = -2..-13: A015371, A015375, A015376, A015377, A015378, A015379, A015380, A015381, A015382, A015384, A015385. - Vincenzo Librandi, Nov 04 2012

Programs

  • Magma
    r:=9; q:=-11; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 04 2012
  • Mathematica
    Table[QBinomial[n, 9, -11],{n, 9, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

Formula

a(n)=product_{i=1..9} ((-11)^(n-i+1)-1)/((-11)^i-1). - Vincenzo Librandi, Nov 04 2012

A015384 Gaussian binomial coefficient [ n,9 ] for q=-12.

Original entry on oeis.org

1, -4762874171, 24747240402737283733, -127616472670861852065241422635, 658504724872263265466971967899949697493, -3397726086395967282512946130260694347212577518123
Offset: 9

Views

Author

Olivier Gérard

Keywords

References

  • 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.

Links

Crossrefs

Cf. Gaussian binomial coefficients [n, 9] for q = -2..-13: A015371, A015375, A015376, A015377, A015378, A015379, A015380, A015381, A015382, A015383, A015385.

Programs

  • Magma
    r:=9; q:=-12; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 04 2012
  • Mathematica
    Table[QBinomial[n, 9, -12],{n, 9, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

Formula

a(n) = Product_{i=1..9} ((-12)^(n-i+1)-1)/((-12)^i-1) (by definition). - Vincenzo Librandi, 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

Views

Author

Olivier Gérard, Dec 11 1999

Keywords

References

  • 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.

Links

Crossrefs

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.

Programs

  • Mathematica
    Table[QBinomial[n, 2, -3], {n, 2, 25}] (* G. C. Greubel, Jul 30 2016 *)
  • PARI
    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
  • Sage
    [gaussian_binomial(n,2,-3) for n in range(2,18)] # Zerinvary Lajos, May 28 2009

Formula

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

Views

Author

Olivier Gérard, Dec 11 1999

Keywords

References

  • 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.

Links

Crossrefs

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).

Programs

  • GAP
    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
  • Magma
    [(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
  • Maple
    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
  • Mathematica
    Table[QBinomial[n, 5, -3], {n, 5, 20}] (* Vincenzo Librandi, Oct 29 2012 *)
  • PARI
    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
  • Sage
    [gaussian_binomial(n,5,-3) for n in range(5,17)] # Zerinvary Lajos, May 27 2009

Formula

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

Views

Author

Olivier Gérard, Dec 11 1999

Keywords

References

  • 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.

Links

Crossrefs

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.

Programs

  • Magma
    [(-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
  • Mathematica
    Table[QBinomial[n, 3, -3], {n, 3, 20}] (* Vincenzo Librandi, Oct 28 2012 *)
  • Maxima
    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 */
  • SageMath
    [gaussian_binomial(n,3,-3) for n in range(3,19)] # Zerinvary Lajos, May 27 2009

Formula

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

Views

Author

Olivier Gérard, Dec 11 1999

Keywords

References

  • 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.

Links

Crossrefs

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).

Programs

  • Mathematica
    Table[QBinomial[n, 6, -3], {n, 6, 20}] (* Vincenzo Librandi, Oct 29 2012 *)
  • Sage
    [gaussian_binomial(n,6,-3) for n in range(6,17)] # Zerinvary Lajos, May 27 2009

Formula

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

Views

Author

Olivier Gérard, Dec 11 1999

Keywords

References

  • 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.

Links

Crossrefs

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).

Programs

  • Mathematica
    Table[QBinomial[n, 7, -3], {n, 7, 20}] (* Vincenzo Librandi, Oct 29 2012 *)
  • Sage
    [gaussian_binomial(n,7,-3) for n in range(7,17)] # Zerinvary Lajos, May 27 2009

Formula

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
Previous Showing 11-18 of 18 results.

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