A015357 Gaussian binomial coefficient [ n,8 ] for q=-3.

1, 4921, 36321901, 229798289941, 1526550040078063, 9974653139743515223, 65533580739687859229563, 429769342296322230713871283, 2820146424148466477944423359046, 18502040831058043147238631145734166

Offset: 8

Author

Olivier Gérard

Keywords

• nonn
• easy

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

• Vincenzo Librandi, Table of n, a(n) for n = 8..200
• Index entries for linear recurrences with constant coefficients, signature (4921,12105660,-8513737740,-2091825362718,169437854380158,4524549298283340,-42209826451809660,-112576695670863081,150094635296999121).

Crossrefs

Cf. Gaussian binomial coefficients [n,8] for q=-2..-13: A015356, A015359, A015360, A015361, A015363, A015364, A015365, A015367, A015368, A015369, A015370. - M. F. Hasler, Nov 03 2012

Gaussian binomial coefficient [n, k]_q for q = -3: A015251 (k = 2), A015268 (k = 3), A015288 (k = 4), A015306 (k = 5), A015324 (k = 6), A015340 (k = 7), this sequence (k = 8), A015375 (k = 9), A015388 (k = 10).

Programs

Magma

r:=8; q:=-3; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 02 2012

Mathematica

Table[QBinomial[n, 8, -3], {n, 8, 20}] (* Vincenzo Librandi, Nov 02 2012 *)

PARI

A015357(n, r=8, q=-3)=prod(i=1, r, (1-q^(n-i+1))/(1-q^i)) \\ M. F. Hasler, Nov 03 2012

Sage

[gaussian_binomial(n,8,-3) for n in range(8,18)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..8} ((-3)^(n-i+1)-1)/((-3)^i-1). - M. F. Hasler, Nov 03 2012

G.f.: -x^8 / ( (x-1)*(27*x+1)*(81*x-1)*(729*x-1)*(9*x-1)*(2187*x+1)*(3*x+1)*(6561*x-1)*(243*x+1) ). - R. J. Mathar, Sep 02 2016

G.f. with offset 0: exp(Sum_{n >= 1} A015518(9*n)/A015518(n) * x^n/n) = 1 + 4921*x + 36321901*x^2 + .... - Peter Bala, Jun 29 2025