A006096 Gaussian binomial coefficient [n, 3] for q = 2.

1, 15, 155, 1395, 11811, 97155, 788035, 6347715, 50955971, 408345795, 3269560515, 26167664835, 209386049731, 1675267338435, 13402854502595, 107225699266755, 857817047249091, 6862582190715075, 54900840777134275, 439207459223777475

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

42*a(n) is a maximum number of intercalates in a Latin square of order 2^n-1 (see A092237). - Eduard I. Vatutin, Apr 24 2025

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.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Links

• T. D. Noe, Table of n, a(n) for n=3..203
• Ronald Orozco López, Generating Functions of Generalized Simplicial Polytopic Numbers and (s,t)-Derivatives of Partial Theta Function, arXiv:2408.08943 [math.CO], 2024. See p. 13.
• Ronald Orozco López, Simplicial d-Polytopic Numbers Defined on Generalized Fibonacci Polynomials, arXiv:2501.11490 [math.CO], 2025. See p. 10.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
• Eduard I. Vatutin, About interconnection between maximum number of intercalates in Latin squares of order N=2^n-1 and Gaussian binomial coefficients [n,3] for q=2 (in Russian).
• Index entries for linear recurrences with constant coefficients, signature (15,-70,120,-64).

Crossrefs

Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), this sequence (k = 3), A006097 (k = 4), A006110 (k = 5), A022189 - A022195 (k = 6 thru 12).

Cf. A092237.

Programs

Magma

r:=3; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 06 2016

Maple

seq((-1+7*2^n-14*4^n+8*8^n)/21,n=1..20);
A006096:=1/(z-1)/(8*z-1)/(2*z-1)/(4*z-1); # Simon Plouffe in his 1992 dissertation with offset 0

Mathematica

Drop[CoefficientList[Series[x^3/((1 - x) (1 - 2 x) (1 - 4 x) (1 - 8 x)), {x, 0, 30}], x], 3]
QBinomial[Range[3,30],3,2] (* Harvey P. Dale, Jan 28 2013 *)

Sage

[gaussian_binomial(n,3,2) for n in range(3,23)] # Zerinvary Lajos, May 24 2009

Formula

G.f.: x^3/((1-x)(1-2x)(1-4x)(1-8x)).

(With a different offset) a(n) = (-1+7*2^n-14*4^n+8*8^n)/21. - James R. Buddenhagen, Dec 14 2003

From Peter Bala, Jul 01 2025: (Start)

a(n) = (q^n - 1)*(q^(n-1) - 1)*(q^(n-2) - 1)/((q^3 - 1)*(q^2 - 1)*(q - 1)) at q = 2.

G.f. with an offset of 0: exp( Sum_{n >= 1} b(4*n)/b(n)*x^n/n ) = 1 + 15*x + 155*x^2 + ..., where b(n) = A000225(n) = 2^n - 1.

The following series telescope:

Sum_{n >= 3} 2^n/(a(n)*a(n+3)) = 420/72075;

Sum_{n >= 3} 4^n/(a(n)*a(n+3)) = 3416/72075;

Sum_{n >= 3} 8^n/(a(n)*a(n+3)) = 28296/72075;

Sum_{n >= 3} 16^n/(a(n)*a(n+3)) = 244748/72075;

Sum_{n >= 3} 32^n/(a(n)*a(n+3)) = 2415315/72075. (End)