A022192 Gaussian binomial coefficients [n, 9] for q = 2.

1, 1023, 698027, 408345795, 222984027123, 117843461817939, 61291693863308051, 31627961868755063955, 16256896431763117598611, 8339787869494479328087443, 4274137206973266943778085267, 2189425218271613769209626653075

Offset: 9

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 9..200

Crossrefs

Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), A006096 (k = 3), A006097 (k = 4), A006110 (k = 5), A022189 - A022195 (k = 6 thru 12).

Programs

Magma

r:=9; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 03 2016

Maple

seq(eval(expand(QDifferenceEquations:-QBinomial(n,9,q)),q=2),n=9..50);

Mathematica

QBinomial[Range[9,20],9,2] (* Harvey P. Dale, Jul 24 2016 *)

PARI

r=9; q=2; for(n=r,30, print1(prod(j=1,r,(1-q^(n-j+1))/(1-q^j)), ", ")) \\ G. C. Greubel, May 30 2018

Sage

[gaussian_binomial(n,9,2) for n in range(9,21)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..9} (2^(n-i+1)-1)/(2^i-1), by definition. - Vincenzo Librandi, Aug 02 2016

G.f.: x^9/Product_{0<=i<=9} (1-2^i*x). - Robert Israel, Apr 23 2017

G.f. with an offset of 0: exp( Sum_{n >= 1} b(10*n)/b(n)*x^n/n ) = 1 + 1023*x + 698027*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - Peter Bala, Jul 01 2025

Extensions

Offset changed by Vincenzo Librandi, Aug 03 2016