A015326 -id:A015326 - cp's OEIS frontend

A015112 Triangle of q-binomial coefficients for q=-4.

1, 1, 1, 1, -3, 1, 1, 13, 13, 1, 1, -51, 221, -51, 1, 1, 205, 3485, 3485, 205, 1, 1, -819, 55965, -219555, 55965, -819, 1, 1, 3277, 894621, 14107485, 14107485, 894621, 3277, 1, 1, -13107, 14317213, -901984419, 3625623645, -901984419, 14317213, -13107, 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 A000012 (k=0), A014985 (k=1), A015253 (k=2), A015271, A015289, A015308, A015326, A015341, A015359, A015376, A015390 (k=10), A015408, A015425,... - M. F. Hasler, Nov 04 2012

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

Flatten[Table[QBinomial[n,m,-4],{n,0,10},{m,0,n}]] (* Harvey P. Dale, Jun 10 2015 *)

T015112(n, k, q=-4)=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

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

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 (52429,916249204,-3695444481856,-3798047410999296,972300137215819776,61999270288106192896,-1007426653738504290304,-3777907597814523756544,4722366482869645213696).

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

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

Table[QBinomial[n, 8, -4], {n, 8, 20}] (* Vincenzo Librandi, Nov 02 2012 *)

A015359(n,r=8,q=-4)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012

[gaussian_binomial(n,8,-4) for n in range(8,16)] # Zerinvary Lajos, May 25 2009

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

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A015112 Triangle of q-binomial coefficients for q=-4.

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

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