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 1-3 of 3 results.

A015116 Triangle of q-binomial coefficients for q=-6.

Original entry on oeis.org

1, 1, 1, 1, -5, 1, 1, 31, 31, 1, 1, -185, 1147, -185, 1, 1, 1111, 41107, 41107, 1111, 1, 1, -6665, 1480963, -8838005, 1480963, -6665, 1, 1, 39991, 53308003, 1910490043, 1910490043, 53308003, 39991, 1, 1, -239945, 1919128099, -412612541285
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), A014987 (k=1), A015257 (k=2), A015273, A015292, A015310, A015328, A015345, A015361, A015378, A015392 (k=10), A015410, A015429,... - M. F. Hasler, Nov 04 2012

Crossrefs

Cf. analog triangles for other q: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), 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
    Table[QBinomial[n, k, -6], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)
  • PARI
    T015116(n, k, q=-6)=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

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

A015308 Gaussian binomial coefficient [ n,5 ] for q = -4.

Original entry on oeis.org

1, -819, 894621, -901984419, 927257668701, -948584595081123, 971588061067577437, -994845394688060798883, 1018737244037427165087837, -1043182954580986851130914723, 1068220365220113899181567068253
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 coefficients [n,5]: A015305 (q=-2), A015306(q=-3), this sequence (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 -205*(-4)^(n-4) +3485*(-4)^(2*n-7) -3485*(-4)^(3*n-9) +205*(-4)^(4*n-10) -(-4)^(5*n-10))/1274203125); # G. C. Greubel, Sep 21 2019
  • Magma
    r:=5; q:=-4; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Aug 03 2016
  • Maple
    seq((1 -205*(-4)^(n-4) +3485*(-4)^(2*n-7) -3485*(-4)^(3*n-9) +205*(-4)^(4*n-10) -(-4)^(5*n-10))/1274203125, n=5..25); # G. C. Greubel, Sep 21 2019
  • Mathematica
    Table[QBinomial[n, 5, -4], {n, 5, 20}] (* Vincenzo Librandi, Oct 29 2012 *)
  • PARI
    a(n) = (1 -205*(-4)^(n-4) +3485*(-4)^(2*n-7) -3485*(-4)^(3*n-9) +205*(-4)^(4*n-10) -(-4)^(5*n-10))/1274203125; \\ G. C. Greubel, Sep 21 2019
  • Sage
    [gaussian_binomial(n,5,-4) for n in range(5,16)] # Zerinvary Lajos, May 27 2009

Formula

a(n) = Product_{i=1..5} ((-4)^(n-i+1)-1)/((-4)^i-1), by definition. - Vincenzo Librandi, Aug 03 2016
G.f.: x^5/((1-x)*(1+4*x)*(1-16*x)*(1+64*x)*(1-256*x)*(1+1024*x)). - R. J. Mathar, Aug 04 2016
From G. C. Greubel, Sep 21 2019: (Start)
a(n) = (1 - 205*(-4)^(n-4) + 3485*(-4)^(2*n-7) - 3485*(-4)^(3*n-9) + 205*(-4)^(4*n-10) - (-4)^(5*n-10))/1274203125.
E.g.f.: exp(-1024*x)*(-1 + 13940*exp(960*x) - 839680*exp(1020*x) + 1048576*exp(1025*x) - 223040*exp(1040*x) + 205*exp(1280*x))/1336098816000000. (End)
Showing 1-3 of 3 results.

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