A006097 Gaussian binomial coefficient [n, 4] for q = 2.

1, 31, 651, 11811, 200787, 3309747, 53743987, 866251507, 13910980083, 222984027123, 3571013994483, 57162391576563, 914807651274739, 14638597687734259, 234230965858250739, 3747802679431278579, 59965700687947706355, 959458073589354016755, 15351384078270441402355

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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 = 4..204
• 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 (31,-310,1240,-1984,1024).

Crossrefs

Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), A006096 (k = 3), this sequence (k = 4), A006110 (k = 5), A022189 - A022195 (k = 6 thru 12).

Programs

Magma

r:=4; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 06 2016

Maple

A006097:=-1/(z-1)/(4*z-1)/(2*z-1)/(8*z-1)/(16*z-1); # Simon Plouffe in his 1992 dissertation with offset 0

Mathematica

faq[n_, q_] = Product[(1-q^(1+k))/(1-q), {k, 0, n-1}];
qbin[n_, m_, q_] = faq[n, q]/(faq[m, q]*faq[n-m, q]);
Table[qbin[n, 4, 2], {n, 4, 21}] (* Jean-François Alcover, Jul 21 2011 *)
QBinomial[Range[4,30],4,2] (* Harvey P. Dale, Dec 10 2012 *)

PARI

a(n)=(2^n-1)*(2^n-2)*(2^n-4)*(2^n-8)/20160 \\ Charles R Greathouse IV, Feb 19 2017

Sage

[gaussian_binomial(n,4,2) for n in range(4,22)] # Zerinvary Lajos, May 24 2009

Formula

G.f.: x^4/((1-x)*(1-2*x)*(1-4*x)*(1-8*x)*(1-16*x)).

a(n) = (2^n-1)*(2^n-2)*(2^n-4)*(2^n-8)/20160. - Bruno Berselli, Aug 29 2011

From Peter Bala, Jul 01 2025: (Start)

G.f. with an offset of 0: exp( Sum_{n >= 1} b(5*n)/b(n)*x^n/n ) = 1 + 31*x + 651*x^2 + ..., where b(n) = A000225(n) = 2^n - 1.

The following series telescope:

Sum_{n >= 4} 2^n/a(n) = 120/7; Sum_{n >= 4} 4^n/a(n) = 2078/7;

Sum_{n >= 4} 8^n/a(n) = 41280/7.

Sum_{n >= 4} 2^n/(a(n)*a(n+4)) = 40/499999;

Sum_{n >= 4} 2^n/(a(n)*a(n+4)*a(n+8)) = 40/6981154678721773;

Sum_{n >= 4} 2^n/(a(n)*a(n+4)*a(n+8)*a(n+12)) = 40/6387876185324781622646124392439. (End)