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.

Showing 1-10 of 18 results. Next

A015370 Gaussian binomial coefficient [ n,8 ] for q=-13.

Original entry on oeis.org

1, 757464241, 621564749363392901, 506798783502833908602716981, 413425812255544017749839936272484623, 337243227617163445881817693983677965955870943, 275099718210633054941121644140453635236773122223471523
Offset: 8

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,8] for q=-2..-12: A015356, A015357, A015359, A015360, A015361, A015363, A015364, A015365, A015367, A015368, A015369. - M. F. Hasler, Nov 03 2012
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), A015385 (r=9), A015402 (r=10), A015422 (r=11), A015438 (r=12). - M. F. Hasler, Nov 03 2012

Programs

  • Magma
    r:=8; q:=-13; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..18]]; // Vincenzo Librandi, Nov 03 2012
  • Mathematica
    Table[QBinomial[n, 8, -13], {n, 8, 14}] (* Vincenzo Librandi, Nov 03 2012 *)
  • PARI
    A015370(n,r=8,q=-13)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012
  • Sage
    [gaussian_binomial(n,8,-13) for n in range(8,14)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..8} ((-13)^(n-i+1)-1)/((-13)^i-1). - M. F. Hasler, Nov 03 2012

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

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, 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).

Programs

  • Magma
    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
  • Mathematica
    Table[QBinomial[n, 9, -3],{n, 9, 20}] (* Vincenzo Librandi, Nov 04 2012 *)
  • Sage
    [gaussian_binomial(n,9,-3) for n in range(9,18)] # Zerinvary Lajos, May 25 2009

Formula

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

Views

Author

Olivier Gérard

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    Flatten[Table[QBinomial[n, m, -3], {n, 0, 50}, {m, 0, n}]] (* Vincenzo Librandi, Nov 01 2012 *)
  • PARI
    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

A015356 Gaussian binomial coefficient [ n,8 ] for q=-2.

Original entry on oeis.org

1, 171, 58311, 13275471, 3624203583, 899790907743, 233988483199263, 59438516325245343, 15275698695588053151, 3902985682508407194271, 1000137219716325891620511, 255910660218571393553843871
Offset: 8

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,8] for q=-3..-13: A015357, A015359, A015360, A015361, A015363, A015364, A015365, A015367, A015368, A015369, A015370. - M. F. Hasler, Nov 03 2012
Diagonal k=8 of the triangular array A015109. See there for further references and programs. - M. F. Hasler, Nov 04 2012

Programs

  • Magma
    r:=8; q:=-2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 02 2012
  • Mathematica
    Table[QBinomial[n, 8, -2], {n, 8, 20}] (* Vincenzo Librandi, Nov 02 2012 *)
  • PARI
    A015356(n, r=8, q=-2)=prod(i=1, r, (q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012
  • Sage
    [gaussian_binomial(n,8,-2) for n in range(8,20)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..8} ((-2)^(n-i+1)-1)/((-2)^i-1). - M. F. Hasler, Nov 03 2012
G.f.: -x^8 / ( (x-1)*(64*x-1)*(128*x+1)*(2*x+1)*(8*x+1)*(32*x+1)*(16*x-1)*(4*x-1)*(256*x-1) ). - R. J. Mathar, Sep 02 2016

A015359 Gaussian binomial coefficient [ n,8 ] for q=-4.

Original entry on oeis.org

1, 52429, 3665049245, 236497451900765, 15559876852907031645, 1018737244037427165087837, 66780267552779682073190144093, 4376244513647234644625387176712285, 286805936690898816904813999400193022045
Offset: 8

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. A015356, A015357, A015360, A015361, A015363, A015364, A015365, A015367 A015368, A015369, A015370 (r=8, q=-2..-13). q=-4 integers/coefficients: A014985 (r=1), A015253 (r=2), A015271 (r=3), A015289 (r=4), A015308 (r=5), A015326 (r=6), A015341 (r=7), A015376 (r=9), A015390 (r=10), A015408 (r=11), A015425 (r=12). - M. F. Hasler, Nov 03 2012

Programs

  • Magma
    r:=8; q:=-4; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..18]]; // Vincenzo Librandi, Nov 03 2012
  • Mathematica
    Table[QBinomial[n, 8, -4], {n, 8, 20}] (* Vincenzo Librandi, Nov 02 2012 *)
  • PARI
    A015359(n,r=8,q=-4)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012
  • Sage
    [gaussian_binomial(n,8,-4) for n in range(8,16)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..8} ((-4)^(n-i+1)-1)/((-4)^i-1). - M. F. Hasler, Nov 03 2012
G.f.: -x^8 / ( (x-1)*(16384*x+1)*(4096*x-1)*(256*x-1)*(65536*x-1)*(64*x+1)*(4*x+1)*(16*x-1)*(1024*x+1) ). - R. J. Mathar, Sep 02 2016

A015360 Gaussian binomial coefficient [ n,8 ] for q=-5.

Original entry on oeis.org

1, 325521, 132454820421, 51329529054158421, 20082729571968536374671, 7842306707330337276457324671, 3063597127265150338968694860387171, 1196702310087594273181943625299134137171, 467463036580276600555969910576099571466559046
Offset: 8

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,8] for q=-2..-13: A015356, A015357, A015359, A015361, A015363, A015364, A015365, A015367, A015368, A015369, A015370. - M. F. Hasler, Nov 03 2012

Programs

  • Magma
    r:=8; q:=-5; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..18]]; // Vincenzo Librandi, Nov 03 2012
  • Mathematica
    Table[QBinomial[n, 8, -5], {n, 8, 20}] (* Vincenzo Librandi, Nov 03 2012 *)
  • PARI
    A015360(n,r=8,q=-5)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012
  • Sage
    [gaussian_binomial(n,8,-5) for n in range(8,16)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..8} ((-5)^(n-i+1)-1)/((-5)^i-1). - M. F. Hasler, Nov 03 2012
G.f.: -x^8 / ( (x-1)*(5*x+1)*(390625*x-1)*(25*x-1)*(625*x-1)*(78125*x+1)*(125*x+1)*(15625*x-1)*(3125*x+1) ). - R. J. Mathar, Sep 02 2016

A015361 Gaussian binomial coefficient [ n,8 ] for q=-6.

Original entry on oeis.org

1, 1439671, 2487182817955, 4158260859792814555, 6989674736616919292088715, 11738459947705882553575280369515, 19716527736890127515275338116221320235, 33116077152651051199781730118147946460139435, 55622326158904300663023790195853299389540017396395
Offset: 8

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,8] for q=-2..-13: A015356, A015357, A015359, A015360, A015363, A015364, A015365, A015367, A015368, A015369, A015370. - M. F. Hasler, Nov 03 2012

Programs

  • Magma
    r:=8; q:=-6; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..18]]; // Vincenzo Librandi, Nov 03 2012
  • Mathematica
    Table[QBinomial[n, 8, -6], {n, 8, 19}] (* Vincenzo Librandi, Nov 03 2012 *)
  • PARI
    A015361(n, r=8, q=-6)=prod(i=1, r, (q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012
  • Sage
    [gaussian_binomial(n,8,-6) for n in range(8,15)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..8} ((-6)^(n-i+1)-1)/((-6)^i-1). - M. F. Hasler, Nov 03 2012
G.f.: -x^8 / ( (x-1)*(279936*x+1)*(216*x+1)*(36*x-1)*(7776*x+1)*(1296*x-1)*(6*x+1)*(46656*x-1)*(1679616*x-1) ). - R. J. Mathar, Sep 02 2016

A015363 Gaussian binomial coefficient [ n,8 ] for q=-7.

Original entry on oeis.org

1, 5044201, 29684623509101, 170628488227082949701, 984049129188697468764456303, 5672509895284807570626050787828903, 32701168672146988445875611556849499108603, 188515500954498588979354521825234382842445990403
Offset: 8

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,8] for q=-2..-13: A015356, A015357, A015359, A015360, A015361, A015364, A015365, A015367, A015368, A015369, A015370. - M. F. Hasler, Nov 03 2012

Programs

  • Magma
    r:=8; q:=-7; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..18]]; // Vincenzo Librandi, Nov 03 2012
  • Mathematica
    QBinomial[Range[8,20],8,-7] (* Harvey P. Dale, May 09 2012 *)
Table[QBinomial[n, 8, -7], {n, 8, 19}] (* Vincenzo Librandi, Nov 03 2012 *)
  • PARI
    A015363(n, r=8, q=-7)=prod(i=1, r, (q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012
  • Sage
    [gaussian_binomial(n,8,-7) for n in range(8,15)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..8} ((-7)^(n-i+1)-1)/((-7)^i-1). - M. F. Hasler, Nov 03 2012

A015364 Gaussian binomial coefficient [ n,8 ] for q=-8.

Original entry on oeis.org

1, 14913081, 254171409198201, 4255976180162154314361, 71420868399845502303592335993, 1198206769685258176958937686297856633, 20102650473193049559156865045854634505718393
Offset: 8

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,8] for q=-2..-13: A015356, A015357, A015359, A015360, A015361, A015363, A015365, A015367, A015368, A015369, A015370. - M. F. Hasler, Nov 03 2012

Programs

  • Magma
    r:=8; q:=-8; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..18]]; // Vincenzo Librandi, Nov 03 2012
  • Mathematica
    Table[QBinomial[n, 8, -8], {n, 8, 15}] (* Vincenzo Librandi, Nov 03 2012 *)
  • PARI
    A015364(n, r=8, q=-8)=prod(i=1, r, (q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012
  • Sage
    [gaussian_binomial(n,8,-8) for n in range(8,15)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..8} ((-8)^(n-i+1)-1)/((-8)^i-1). - M. F. Hasler, Nov 03 2012

A015365 Gaussian binomial coefficient [ n,8 ] for q=-9.

Original entry on oeis.org

1, 38742049, 1688564650965445, 72587599955185580267365, 3125134483161392104770081009295, 134524513999723596604019036560420619887, 5790850118312580284352508983888376537699322083
Offset: 8

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,8] for q=-2..-13: A015356, A015357, A015359, A015360, A015361, A015363, A015364, A015367, A015368, A015369, A015370. - M. F. Hasler, Nov 03 2012

Programs

  • Magma
    r:=8; q:=-9; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..18]]; // Vincenzo Librandi, Nov 03 2012
  • Mathematica
    Table[QBinomial[n, 8, -9], {n, 8, 15}] (* Vincenzo Librandi, Nov 03 2012 *)
  • PARI
    A015365(n, r=8, q=-9)=prod(i=1, r, (q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012
  • Sage
    [gaussian_binomial(n,8,-9) for n in range(8,14)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..8} ((-9)^(n-i+1)-1)/((-9)^i-1). - M. F. Hasler, Nov 03 2012
Showing 1-10 of 18 results. Next

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