OFFSET
3,2
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).
FORMULA
G.f.: x^3/((1-x)*(1-2*x)*(1-4*x)*(1-8*x)).
(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)
From Enrique Navarrete, Apr 06 2026: (Start)
a(n) = (8^n - 7*4^n + 14*2^n - 8)/168.
a(n) = 15*a(n-1) - 70*a(n-2) + 120*a(n-3) - 64*a(n-4).
E.g.f.: (1/168)*exp(x)*(exp(7*x) - 7*exp(3*x) + 14*exp(x) - 8). (End)
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 *)
PROG
(SageMath) [gaussian_binomial(n, 3, 2) for n in range(3, 23)] # Zerinvary Lajos, May 24 2009
(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
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved