A006112 Gaussian binomial coefficient [ n,3 ] for q = 5. (Formerly M5404)

1, 156, 20306, 2558556, 320327931, 40053706056, 5007031143556, 625886840206056, 78236053707784181, 9779511680526143556, 1222439084242108174806, 152804888634672088643556, 19100611156944225555440431, 2387576396558283557830831056

Offset

3,2

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).

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

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

Mathematica

Table[QBinomial[n, 3, 5], {n, 3, 20}] (* Vincenzo Librandi, Aug 07 2016 *)

Prog

(Sage) [gaussian_binomial(n, 3, 5) for n in range(3, 14)] # Zerinvary Lajos, May 27 2009
(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

Crossrefs

Sequence in context: A035838 A035824 A252778 * A239410 A216788 A239338

Adjacent sequences: A006109 A006110 A006111 * A006113 A006114 A006115

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved