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 19 results. Next

A015402 Gaussian binomial coefficient [ n,10 ] for q=-13.

Original entry on oeis.org

1, 128011456717, 17752510805031727164870, 2446220929187500105890055171302510, 337244135881870906696294510219932684378716373, 46491842741544248966048667175076748587505712393943779761
Offset: 10

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,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

Programs

  • Magma
    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
  • Mathematica
    Table[QBinomial[n, 10, -13], {n, 10, 20}] (* Vincenzo Librandi, Nov 05 2012 *)
  • PARI
    A015402(n,r=10,q=-13)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012
  • Sage
    [gaussian_binomial(n,10,-13) for n in range(10,15)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..10} ((-13)^(n-i+1)-1)/((-13)^i-1). - M. F. Hasler, Nov 03 2012

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

Original entry on oeis.org

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

Programs

  • Magma
    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
  • Mathematica
    Table[QBinomial[n, 8, -3], {n, 8, 20}] (* Vincenzo Librandi, Nov 02 2012 *)
  • PARI
    A015357(n, r=8, q=-3)=prod(i=1, r, (1-q^(n-i+1))/(1-q^i)) \\ M. F. Hasler, Nov 03 2012
  • Sage
    [gaussian_binomial(n,8,-3) for n in range(8,18)] # Zerinvary Lajos, May 25 2009

Formula

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

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

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

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, 10] for q = -2..-13: A015386, A015388, A015391, A015392, A015393, A015394, A015397, A015398, A015399, A015401, A015402. - Vincenzo Librandi, Nov 04 2012

Programs

  • Magma
    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
  • Mathematica
    Table[QBinomial[n, 10, -4], {n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)
  • Sage
    [gaussian_binomial(n,10,-4) for n in range(10,17)] # Zerinvary Lajos, May 25 2009

Formula

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

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, 10] for q = -2..-13: A015386, A015388, A015390, A015392, A015393, A015394, A015397, A015398, A015399, A015401, A015402.

Programs

  • Magma
    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
  • Mathematica
    Table[QBinomial[n, 10, -5], {n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)
  • Sage
    [gaussian_binomial(n,10,-5) for n in range(10,17)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..10} ((-5)^(n-i+1)-1)/((-5)^i-1) (by definition). - Vincenzo Librandi, Nov 04 2012

A015386 Gaussian binomial coefficient [ n,10 ] for q=-2.

Original entry on oeis.org

1, 683, 932295, 848699215, 926949282623, 920460637644639, 957498220445101855, 972884994173649887135, 1000137219716325891620511, 1022146087305755916943130783, 1047699739488399814866709052575, 1072321450350081081965428740719775
Offset: 10

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, 10] for q = -2..-13: A015388, A015390, A015391, A015392, A015393, A015394, A015397, A015398, A015399, A015401, A015402.

Programs

  • Magma
    r:=10; q:=-2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 04 2012
  • Mathematica
    Table[QBinomial[n, 10, -2],{n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)
  • Sage
    [gaussian_binomial(n,10,-2) for n in range(10,21)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..10} ((-2)^(n-i+1)-1)/((-2)^i-1) (by definition). - Vincenzo Librandi, Nov 04 2012
G.f.: x^10 / ( (x-1)*(512*x+1)*(64*x-1)*(128*x+1)*(1024*x-1)*(2*x+1)*(8*x+1)*(32*x+1)*(16*x-1)*(4*x-1)*(256*x-1) ). - R. J. Mathar, Sep 22 2016

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

Original entry on oeis.org

1, 51828151, 3223388672928931, 194007802557550502202331, 11739968552378570066280405695371, 709779726467093092873777345973423761771, 42918585756017923252384776090351752769462732331
Offset: 10

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, 10] for q = -2..-13: A015386, A015388, A015390, A015391, A015393, A015394, A015397, A015398, A015399, A015401, A015402.

Programs

  • Magma
    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
  • Mathematica
    Table[QBinomial[n, 10, -6], {n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)
  • Sage
    [gaussian_binomial(n,10,-6) for n in range(10,16)] # Zerinvary Lajos, May 25 2009

Formula

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

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, 10] for q = -2..-13: A015386, A015388, A015390, A015391, A015392, A015394, A015397, A015398, A015399, A015401, A015402. - Vincenzo Librandi, Nov 04 2012

Programs

  • Magma
    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
  • Mathematica
    Table[QBinomial[n, 10, -7], {n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)
  • Sage
    [gaussian_binomial(n,10,-7) for n in range(10,16)] # Zerinvary Lajos, May 25 2009

Formula

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

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, 10] for q = -2..-13: A015386, A015388, A015390, A015391, A015392, A015393, A015397, A015398, A015399, A015401, A015402. - Vincenzo Librandi, Nov 04 2012

Programs

  • Magma
    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
  • Mathematica
    Table[QBinomial[n, 10, -8], {n, 10, 20}] (* Vincenzo Librandi, Nov 04 2012 *)
  • Sage
    [gaussian_binomial(n,10,-8) for n in range(10,16)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..10} ((-8)^(n-i+1)-1)/((-8)^i-1) (by definition). - Vincenzo Librandi, Nov 04 2012
Showing 1-10 of 19 results. Next

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