A006112 Gaussian binomial coefficient [ n,3 ] for q = 5.
1, 156, 20306, 2558556, 320327931, 40053706056, 5007031143556, 625886840206056, 78236053707784181, 9779511680526143556, 1222439084242108174806, 152804888634672088643556, 19100611156944225555440431, 2387576396558283557830831056
Offset: 3
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
- Vincenzo Librandi, Table of n, a(n) for n = 3..200
- 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)
- Index entries for linear recurrences with constant coefficients, signature (156, -4030, 19500, -15625).
Programs
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Magma
r:=3; q:=5; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 07 2016
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Maple
A006112:=1/(z-1)/(125*z-1)/(25*z-1)/(5*z-1); # [conjectured by Simon Plouffe in his 1992 dissertation] seq((5^n-1)*(5^n-5)*(5^n-25)/1488000, n=3..30); # Robert Israel, Feb 01 2018
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Mathematica
Table[QBinomial[n, 3, 5], {n, 3, 20}] (* Vincenzo Librandi, Aug 07 2016 *) -
Sage
[gaussian_binomial(n,3,5) for n in range(3,14)] # Zerinvary Lajos, May 27 2009
Formula
G.f.: x^3/((1-x)*(1-5*x)*(1-25*x)*(1-125*x)). - Vincenzo Librandi, Aug 07 2016
a(n) = Product_{i=1..3} (5^(n-i+1)-1)/(5^i-1), by definition. - Vincenzo Librandi, Aug 07 2016
a(n) = (5^n-1)*(5^n-5)*(5^n-25)/1488000. - Robert Israel, Feb 01 2018