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A015109 Triangle of Gaussian (or q-binomial) coefficients for q=-2. 30
1, 1, 1, 1, -1, 1, 1, 3, 3, 1, 1, -5, 15, -5, 1, 1, 11, 55, 55, 11, 1, 1, -21, 231, -385, 231, -21, 1, 1, 43, 903, 3311, 3311, 903, 43, 1, 1, -85, 3655, -25585, 56287, -25585, 3655, -85, 1, 1, 171, 14535, 208335, 875007, 875007, 208335, 14535, 171, 1, 1, -341, 58311 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

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), A077925 (k=1), A015249 (k=2), A015266 (k=3), A015287 (k=4), A015305 (k=5), A015323 (k=6), A015338 (k=7), A015356 (k=8), A015371 (k=9), A015386 (k=10), A015405 (k=11), A015423 (k=12), ... - M. F. Hasler, Nov 04 2012

The elements of the inverse matrix are apparently T^(-1)(n,k) = (-1)^n*A157785(n,k) - R. J. Mathar, Mar 12 2013

Fu et al. give two combinatorial interpretations of the (unsigned) q-binomial coefficients when q is a negative integer. - Peter Bala, Nov 02 2017

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

J. A. de Azcarraga and J. A. Macfarlane, Group Theoretical Foundations of Fractional Supersymmetry, arxiv:hep-th/9506177 (1995).

S. Fu, V. Reiner, D. Stanton and N. Thiem, The negative q-binomial, arXiv:1108.4702v1 [math.CO]

R. Parthasarathy, q-Fermionic Numbers and Their Roles in Some Physical Problems, arxiv:quant-ph/0403216, 2004.

FORMULA

From Roger L. Bagula, Feb 10 2009: (Start)

t(n,m) = n! if m = 0, otherwise Product_{k=1..n} Sum_{i=0..k-1} -(m + 1)^i;

C(n,k,m) = 1 if n = 0, otherwise t(n, m)/(t(k, m)*t(n - k, m)). (End)

EXAMPLE

From Roger L. Bagula, Feb 10 2009: (Start)

1;

1, 1;

1, -1, 1;

1, 3, 3, 1;

1, -5, 15, -5, 1;

1, 11, 55, 55, 11, 1;

1, -21, 231, -385, 231, -21, 1;

1, 43, 903, 3311, 3311, 903, 43, 1;

1, -85, 3655, -25585, 56287, -25585, 3655, -85, 1;

1, 171, 14535, 208335, 875007, 875007, 208335, 14535, 171, 1;

1, -341, 58311, -1652145, 14208447, -27125217, 14208447, -1652145, 58311, -341, 1;

(...) (End)

MAPLE

A015109 := proc(n, k)

   mul( ((-2)^(1+n-i)-1)/((-2)^i-1) , i=1..k) ;

end proc: # R. J. Mathar, Mar 12 2013

MATHEMATICA

t[n_, m_] = If[m == 0, n!, Product[Sum[(-(m + 1))^i, {i, 0, k - 1}], {k, 1, n}]];

b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];

c = Table[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];

TableForm[c];

Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]

(* Roger L. Bagula, Feb 10 2009 *)

Table[QBinomial[n, k, -2], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)

PROG

(PARI) T015109(n, k, q=-2)=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

CROSSREFS

Cf. A015152 (row sums).

Cf. A022166 (q=2), A022167, 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.

Analogous triangles for other q: 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), A015132 (q=-14), A015133 (q=-15).

Sequence in context: A196989 A034871 A333758 * A319699 A157636 A086626

Adjacent sequences:  A015106 A015107 A015108 * A015110 A015111 A015112

KEYWORD

sign,tabl,easy

AUTHOR

Olivier Gérard

EXTENSIONS

Edited by M. F. Hasler, Nov 04 2012

STATUS

approved

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Last modified July 2 12:56 EDT 2020. Contains 335400 sequences. (Running on oeis4.)