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%I #17 Feb 22 2022 03:37:25
%S -1,6,105,440,1155,2394,4301,7020,10695,15470,21489,28896,37835,48450,
%T 60885,75284,91791,110550,131705,155400,181779,210986,243165,278460,
%U 317015,358974,404481,453680,506715
%N a(n) = (2*n-1)*(3*n-1)*(4*n-1).
%H G. C. Greubel, <a href="/A033589/b033589.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Harvey P. Dale_, Sep 22 2014
%F G.f.: (-1 +10*x +75*x^2 +60*x^3)/(1-x)^4. - _R. J. Mathar_, Feb 06 2017
%F From _G. C. Greubel_, Mar 05 2020: (Start)
%F a(n) = n^3 * Pochhammer(2 - 1/n, 3) = Product_{j=2..4} (j*n-1).
%F E.g.f.: (-1 + 7*x + 46*x^2 + 24*x^3)*exp(x). (End)
%F Sum_{n>=1} 1/a(n) = (sqrt(3)/2-1)*Pi + 8*log(2) - 9*log(3)/2. - _Amiram Eldar_, Feb 22 2022
%p seq( mul(j*n-1, j=2..4), n=0..30); # _G. C. Greubel_, Mar 05 2020
%t Table[Times@@(n*Range[2,4]-1),{n,0,30}] (* or *) LinearRecurrence[{4,-6,4,-1},{-1,6,105,440},30] (* _Harvey P. Dale_, Sep 22 2014 *)
%o (PARI) vector(31, n, my(m=n-1); prod(j=2,4, j*m-1) ) \\ _G. C. Greubel_, Mar 05 2020
%o (Magma) [(2*n-1)*(3*n-1)*(4*n-1): n in [0..30]]; // _G. C. Greubel_, Mar 05 2020
%o (Sage) [product(j*n-1 for j in (2..4)) for n in (0..30)] # _G. C. Greubel_, Mar 05 2020
%Y Cf. A060747, A033568, A033590.
%K sign,easy
%O 0,2
%A _N. J. A. Sloane_
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