A015309 -id:A015309 - cp's OEIS frontend

A015113 Triangle of q-binomial coefficients for q=-5.

1, 1, 1, 1, -4, 1, 1, 21, 21, 1, 1, -104, 546, -104, 1, 1, 521, 13546, 13546, 521, 1, 1, -2604, 339171, -1679704, 339171, -2604, 1, 1, 13021, 8476671, 210302171, 210302171, 8476671, 13021, 1, 1, -65104, 211929796, -26279294704, 131649159046

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), A014986 (k=1), A015255 (k=2), A015272, A015291, A015309, A015327, A015344, A015360, A015377, A015391 (k=10), A015409, A015427,... - M. F. Hasler, Nov 04 2012

Cf. analog triangles for other q: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), 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

Table[QBinomial[n, k, -5], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)

T015113(n, k, q=-5)=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

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 for linear recurrences with constant coefficients, signature (-182,16653,428220,-4046679,-10746918,14348907)

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).

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

[(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

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

Table[QBinomial[n, 5, -3], {n, 5, 20}] (* Vincenzo Librandi, Oct 29 2012 *)

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

[gaussian_binomial(n,5,-3) for n in range(5,17)] # Zerinvary Lajos, May 27 2009

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

A071952 Diagonal T(n, 4) of triangle in A071951.

Original entry on oeis.org

1, 40, 1092, 25664, 561104, 11807616, 243248704, 4950550528, 100040447232, 2013177300992, 40412056994816, 810023815790592, 16221871691714560, 324694197936160768, 6496965245491888128, 129976281056339296256

Offset: 4

N. J. A. Sloane, Jun 16 2002

G. C. Greubel, Table of n, a(n) for n = 4..250
W. N. Everitt, L. L. Littlejohn and R. Wellman, Legendre polynomials, Legendre-Stirling numbers and the left-definite spectral analysis of the Legendre differential expression, J. Comput. Appl. Math. 148, 2002, 213-238.
L. L. Littlejohn and R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181(2), 2002, 280-339.
Index entries for linear recurrences with constant coefficients, signature (40,-508,2304,-2880).

Cf. A000079, A016129, A015309, A089278, A089500.

Cf. A071951, A129467.

List([4..20], n-> 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315); # G. C. Greubel, Mar 16 2019

[2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315: n in [4..20]]; // G. C. Greubel, Mar 16 2019

Flatten[ Table[ Sum[(-1)^{r + 4}(2r + 1)(r^2 + r)^n/((r + 5)!(4 - r)!), {r, 1, 4}], {n, 4, 20}]]
LinearRecurrence[{40, -508, 2304, -2880}, {1, 40, 1092, 25664}, 20] (* G. C. Greubel, Mar 16 2019 *)

{a(n) = 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315}; \\ G. C. Greubel, Mar 16 2019

[2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315 for n in (4..20)] # G. C. Greubel, Mar 16 2019

From Wolfdieter Lang, Nov 07 2003: (Start)
a(n+4) = A071951(n+4, 4) = (-7*2^n + 405*6^n - 2268*12^n + 2500*20^n)/630, n >= 0.
G.f.: x^4/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)). (End)
a(n) = det(|ps(i+2,j+1)|, 1 <= i,j <= n-4), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467) and n > 3. - Mircea Merca, Apr 06 2013
From G. C. Greubel, Mar 16 2019: (Start)
a(n) = 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315.
E.g.f.: (1 - exp(2*x))^4*(14 + 28*exp(2*x) + 28*exp(4*x) + 20*exp(6*x) + 10*exp(8*x) + 4*exp(10*x) + exp(12*x))/8!. (End)

More terms from Robert G. Wilson v, Jun 19 2002

Definition corrected by Georg Fischer, Jul 07 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

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 for linear recurrences with constant coefficients, signature (-819,223860,14051520,-229232640,-858783744,1073741824).

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).

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

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

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

Table[QBinomial[n, 5, -4], {n, 5, 20}] (* Vincenzo Librandi, Oct 29 2012 *)

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

[gaussian_binomial(n,5,-4) for n in range(5,16)] # Zerinvary Lajos, May 27 2009

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)

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A015113 Triangle of q-binomial coefficients for q=-5.

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A015306 Gaussian binomial coefficient [ n,5 ] for q = -3.

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A071952 Diagonal T(n, 4) of triangle in A071951.

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

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