A015109 -id:A015109 - cp's OEIS frontend

A015121 Triangle of q-binomial coefficients for q=-9.

1, 1, 1, 1, -8, 1, 1, 73, 73, 1, 1, -656, 5986, -656, 1, 1, 5905, 484210, 484210, 5905, 1, 1, -53144, 39226915, -352504880, 39226915, -53144, 1, 1, 478297, 3177326971, 257015284435, 257015284435, 3177326971, 478297, 1, 1, -4304672, 257363962948

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), A014991 (k=1), A015260 (k=2), A015277 (k=3), A015295 (k=4), A015315 (k=5), A015332 (k=6), A015349 (k=7), A015365 (k=8), A015381 (k=9), A015397 (k=10), A015414 (k=11), A015432 (k=12). - M. F. Hasler, Nov 05 2012

Index entries related to Gaussian binomial coefficients.

Cf. analog triangles for other q: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015118 (q=-8), A015123 (q=-10), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15). - M. F. Hasler, Nov 04 2012

Cf. analog triangles for positive q=2,...,24: 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 05 2012

Table[QBinomial[n, k, -9], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)

T015121(n, k, q=-9)=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

A015123 Triangle of q-binomial coefficients for q=-10.

Original entry on oeis.org

1, 1, 1, 1, -9, 1, 1, 91, 91, 1, 1, -909, 9191, -909, 1, 1, 9091, 918191, 918191, 9091, 1, 1, -90909, 91828191, -917272809, 91828191, -90909, 1, 1, 909091, 9182728191, 917364637191, 917364637191, 9182728191, 909091, 1, 1, -9090909, 918273728191

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 in the former, or row/columns in the latter, are then (k=0,...,12): A000012, A014992, A015261, A015278, A015298, A015316, A015333, A015350, A015367, A015382, A015398, A015417, A015433. - M. F. Hasler, Nov 04 & Nov 05 2012

Index entries related to Gaussian binomial coefficients.

Cf. analog triangles for other q: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015118 (q=-8), A015121 (q=-9), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15). - M. F. Hasler, Nov 04 2012

Cf. analog triangles for positive q=2,...,24: 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 05 2012

Table[QBinomial[n, k, -10], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)

T015123(n, k, q=-10)=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

A015124 Triangle of q-binomial coefficients for q=-11.

Original entry on oeis.org

1, 1, 1, 1, -10, 1, 1, 111, 111, 1, 1, -1220, 13542, -1220, 1, 1, 13421, 1637362, 1637362, 13421, 1, 1, -147630, 198134223, -2177691460, 198134223, -147630, 1, 1, 1623931, 23974093353, 2898705467483, 2898705467483, 23974093353, 1623931, 1, 1

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 in the former, or row/columns in the latter, are then (k=0,...,12): A000012, A014993, A015262, A015279, A015300, A015317, A015334, A015353, A015368, A015383, A015499, A015418, A015434. - M. F. Hasler, Nov 04 & Nov 05 2012

Index entries related to Gaussian binomial coefficients.

Cf. analog triangles for other negative q=-2,...,-15: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015118 (q=-8), A015121 (q=-9), A015123 (q=-10), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15). - M. F. Hasler, Nov 04 2012

Cf. analog triangles for positive q=2,...,24: 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 05 2012

T015124(n, k, q=-11)=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

A015125 Triangle of q-binomial coefficients for q=-12.

Original entry on oeis.org

1, 1, 1, 1, -11, 1, 1, 133, 133, 1, 1, -1595, 19285, -1595, 1, 1, 19141, 2775445, 2775445, 19141, 1, 1, -229691, 399683221, -4793193515, 399683221, -229691, 1, 1, 2756293, 57554154133, 8283038077141, 8283038077141, 57554154133, 2756293, 1, 1

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, for k=0,...,12: A000012, A014994, A015264, A015281, A015302, A015319, A015336, A015354, A015369, A015384, A015401, A015421, A015436. - M. F. Hasler, Nov 04 2012

Index entries related to Gaussian binomial coefficients.

Cf. analog triangles for other negative q=-2,...,-15: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015118 (q=-8), A015121 (q=-9), A015123 (q=-10), A015124 (q=-11), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15). - M. F. Hasler, Nov 04 2012

Cf. analog triangles for positive q=2,...,24: 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 05 2012

T015125(n, k, q=-12)=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

A015132 Triangle of (Gaussian) q-binomial coefficients for q=-14.

Original entry on oeis.org

1, 1, 1, 1, -13, 1, 1, 183, 183, 1, 1, -2561, 36051, -2561, 1, 1, 35855, 7063435, 7063435, 35855, 1, 1, -501969, 1384469115, -19375002205, 1384469115, -501969, 1, 1, 7027567, 271355444571, 53166390519635, 53166390519635, 271355444571

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). - M. F. Hasler, Nov 04 2012

Index entries related to Gaussian binomial coefficients.

Cf. analog triangles for other negative q=-2,...,-15: A015109 (q=-2), A015110 (q=-3), 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), A015133 (q=-15). - M. F. Hasler, Nov 04 2012

Cf. analog triangles for positive q=2,...,24: 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 05 2012

T015132(n, k, q=-14)=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

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 (171,29070,-1666680,-56000448,896007168,6826721280,-30482104320,-45902462976,68719476736).

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

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

Table[QBinomial[n, 8, -2], {n, 8, 20}] (* Vincenzo Librandi, Nov 02 2012 *)

A015356(n, r=8, q=-2)=prod(i=1, r, (q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012

[gaussian_binomial(n,8,-2) for n in range(8,20)] # Zerinvary Lajos, May 25 2009

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

A015371 Gaussian binomial coefficient [ n,9 ] for q=-2.

Original entry on oeis.org

1, -341, 232903, -105970865, 57881286463, -28735427761313, 14946527496991519, -7593183562134412385, 3902985682508407194271, -1994425683761796076272481, 1022146087305755916943130783, -523082886040328458081329117025

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

Diagonal k=9 of the triangular array A015109. See there for further references and programs. - M. F. Hasler, Nov 04 2012

Cf. Gaussian binomial coefficients [n,9] for q=-2..-13: A015375, A015376, A015377, A015378, A015379, A015380, A015381, A015382, A015383, A015384, A015385. - Vincenzo Librandi, Nov 04 2012

r:=9; q:=-2; [&*[(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, -2],{n, 9, 20}] (* Vincenzo Librandi, Nov 04 2012 *)

[gaussian_binomial(n,9,-2) for n in range(9,21)] # Zerinvary Lajos, May 25 2009

a(n) = Product_{i=1..9} ((-2)^(n-i+1)-1)/((-2)^i-1). - Vincenzo Librandi, Nov 04 2012

G.f.: -x^9 / ( (x-1)*(512*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

A015305 Gaussian binomial coefficient [ n,5 ] for q = -2.

Original entry on oeis.org

1, -21, 903, -25585, 875007, -27125217, 882215391, -28005209505, 899790907743, -28735427761313, 920460637644639, -29439916001972385, 942314556807454559, -30150270336284213409, 964869381941043396447, -30874848551033891160225

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 related to Gaussian binomial coefficients.
Index entries for linear recurrences with constant coefficients, signature (-21,462,3080,-14784,-21504,32768).

Diagonal k=5 of the triangular array A015109. See there for further references and programs. - M. F. Hasler, Nov 04 2012

List([5..25], n-> (1 -11*(-2)^(n-4) +55*(-2)^(2*n-7) -55*(-2)^(3*n-9) +11*(-2)^(4*n- 10) -(-2)^(5*n-10))/40095 ); # G. C. Greubel, Sep 21 2019

[(1 -11*(-2)^(n-4) +55*(-2)^(2*n-7) -55*(-2)^(3*n-9) +11*(-2)^(4*n-10) -(-2)^(5*n-10))/40095: n in [5..25]]; // G. C. Greubel, Sep 21 2019

seq((1 -11*(-2)^(n-4) +55*(-2)^(2*n-7) -55*(-2)^(3*n-9) +11*(-2)^(4*n- 10) -(-2)^(5*n-10))/40095, n=5..25); # G. C. Greubel, Sep 21 2019

Table[QBinomial[n, 5, -2], {n, 5, 20}] (* Vincenzo Librandi, Oct 29 2012 *)

a(n) = (1 -11*(-2)^(n-4) +55*(-2)^(2*n-7) -55*(-2)^(3*n-9) +11*(-2)^(4*n- 10) -(-2)^(5*n-10))/40095 \\ G. C. Greubel, Sep 21 2019

[gaussian_binomial(n,5,-2) for n in range(5,21)] # Zerinvary Lajos, May 27 2009

A015305(n) = T[n,5], where T is the triangular array A015109. - M. F. Hasler, Nov 04 2012
G.f.: x^5/((1-x)*(1+2*x)*(1-4*x)*(1+8*x)*(1-16*x)*(1+32*x)). - R. J. Mathar, Aug 03 2016
From G. C. Greubel, Sep 21 2019: (Start)
a(n) = (1 -11*(-2)^(n-4) +55*(-2)^(2*n-7) -55*(-2)^(3*n-9) +11*(-2)^(4*n- 10) -(-2)^(5*n-10))/40095.
E.g.f.: (11*exp(16*x) - 440 + 1024*exp(x) - 704*exp(-2*x) + 110*exp(-8*x) - exp(-32*x))/41057280. (End)

A015266 Gaussian binomial coefficient [ n,3 ] for q = -2.

Original entry on oeis.org

1, -5, 55, -385, 3311, -25585, 208335, -1652145, 13275471, -105970865, 848699215, -6785865905, 54301841231, -434355079345, 3475079247695, -27799679551665, 222401254176591, -1779194762447025, 14233619183613775

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..1000
Index entries for linear recurrences with constant coefficients, signature (-5, 30, 40, -64).

Diagonal k=3 of the triangular array A015109. See there for further references and programs. - M. F. Hasler, Nov 04 2012

[(1/81)*(24*4^n-6*(-2)^n+64*(-8)^n-1): n in [0..20]]; // Vincenzo Librandi, Aug 23 2011

Table[QBinomial[n, 2, -2], {n, 3, 25}] (* G. C. Greubel, Jul 31 2016 *)

[gaussian_binomial(n,3,-2) for n in range(3,22)] # Zerinvary Lajos, May 27 2009

From Paul Barry, Jul 12 2005: (Start)
G.f.: x^3/((1-2*x-8*x^2)*(1+7*x-8*x^2));
a(n) = -5*a(n-1) + 30*a(n-2) + 40*a(n-3) - 64*a(n-4);
a(n+3) = (-1)^n*J(n)*J(n+1)*J(n+2)/3, where J(n)=A001045(n). (End)
a(n) = T015109(n,3), where T015109 is the triangular array defined by A015109. - M. F. Hasler, Nov 04 2012

A015287 Gaussian binomial coefficient [ n,4 ] for q = -2.

Original entry on oeis.org

1, 11, 231, 3311, 56287, 875007, 14208447, 225683007, 3624203583, 57881286463, 926949282623, 14824402656063, 237244744338239, 3795481554332479, 60731179948567359, 971671079497526079, 15546959673214593855

Offset: 4

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 = 4..800
Index entries for linear recurrences with constant coefficients, signature (11,110,-440,-704,1024).
Index entries related to Gaussian binomial coefficients.

Diagonal k=4 in the triangular array A015109. See there for further references and programs. - M. F. Hasler, Nov 04 2012

r:=4; q:=-2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 02 2016

Table[QBinomial[n, 4, -2], {n, 4, 20}] (* Vincenzo Librandi, Oct 28 2012 *)

[gaussian_binomial(n,4,-2) for n in range(4,21)] # Zerinvary Lajos, May 27 2009

G.f.: x^4/((1-x)*(1+2*x)*(1-4*x)*(1+8*x)*(1-16*x)). - Bruno Berselli, Oct 30 2012
a(n) = (1 - 2^(2n-5)*(15-2^(2n-1)) - (-1)^n*5*2^(n-3)*(1-2^(2n-3)))/1215. - Bruno Berselli, Oct 30 2012
A015287(n) = T[n,4], where T is the triangular array A015109. - M. F. Hasler, Nov 04 2012

cp's OEIS Frontend

A015121 Triangle of q-binomial coefficients for q=-9.

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A015123 Triangle of q-binomial coefficients for q=-10.

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A015124 Triangle of q-binomial coefficients for q=-11.

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A015125 Triangle of q-binomial coefficients for q=-12.

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A015132 Triangle of (Gaussian) q-binomial coefficients for q=-14.

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

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A015371 Gaussian binomial coefficient [ n,9 ] for q=-2.

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A015305 Gaussian binomial coefficient [ n,5 ] for q = -2.

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