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A022231 Gaussian binomial coefficients [ n,2 ] for q = 7.

%I A022231 #34 Nov 04 2023 12:03:50
%S A022231 1,57,2850,140050,6865251,336416907,16484565700,807744680100,
%T A022231 39579496050501,1939395353553757,95030372653688550,
%U A022231 4656488262337620150,228167924870691555751,11180228318776923410607,547831187620860507371400
%N A022231 Gaussian binomial coefficients [ n,2 ] for q = 7.
%H A022231 Vincenzo Librandi, <a href="/A022231/b022231.txt">Table of n, a(n) for n = 2..200</a> (first 591 terms from Todd Silvestri)
%H A022231 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (57, -399, 343).
%F A022231 G.f.: x^2/((1-x)*(1-7*x)*(1-49*x)).
%F A022231 a(n) = (7^(n+1)-1)*(7^(n+2)-1)/288. - _Todd Silvestri_, Dec 16 2014
%F A022231 E.g.f.: (343*exp(49*x)-56*exp(7*x)+exp(x))/288. - _Robert Israel_, Dec 16 2014
%F A022231 a(n+3) = 57*a(n+2) - 399*a(n+1) + 343*a(n). - _Robert Israel_, Dec 16 2014
%p A022231 seq((7^(n+1)-1)*(7^(n+2)-1)/288, n=0..30); # _Robert Israel_, Dec 16 2014
%t A022231 a[n_Integer/;n>=0]:=(7^(n+1)-1)*(7^(n+2)-1)/288 (* _Todd Silvestri_, Dec 16 2014 *)
%t A022231 Table[QBinomial[n, 2, 7], {n, 2, 20}] (* _Vincenzo Librandi_, Aug 12 2016 *)
%o A022231 (Sage) [gaussian_binomial(n,2,7) for n in range(2,17)] # _Zerinvary Lajos_, May 28 2009
%o A022231 (PARI) Vec(1/((1-x)*(1-7*x)*(1-49*x)) + O(x^30)) \\ _Michel Marcus_, Dec 16 2014
%o A022231 (PARI) r=2; q=7; for(n=r,30, print1(prod(j=1,r,(1-q^(n-j+1))/(1-q^j)), ", ")) \\ _G. C. Greubel_, Jun 13 2018
%o A022231 (PARI) lista(nn, na=2, q=7) = qp=matpascal(nn+q, q); vector(nn, n, qp[n+na, n]); \\ _Michel Marcus_, Jun 13 2018
%o A022231 (Magma) r:=2; q:=7; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // _Vincenzo Librandi_, Aug 12 2016
%K A022231 nonn,easy
%O A022231 2,2
%A A022231 _N. J. A. Sloane_
%E A022231 Offset changed by _Vincenzo Librandi_, Aug 12 2016