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%I #36 Mar 11 2022 07:30:33
%S 1,0,105,880,3465,9576,21505,42120,74865,123760,193401,288960,416185,
%T 581400,791505,1053976,1376865,1768800,2238985,2797200,3453801,
%U 4219720,5106465,6126120,7291345,8615376,10112025,11795680,13681305,15784440,18121201,20708280,23562945
%N a(n) = (n-1)*(2*n-1)*(3*n-1)*(4*n-1).
%C The sequence of n such that n is prime and (2*n+1) is prime is the sequence of Sophie Germain primes A005384; the subsequence of those for which in addition (3*n+2) is prime is A067256; and the subsequence of those for which in addition (4*n+3) is prime is A067257. - _Jonathan Vos Post_, Dec 15 2004
%H G. C. Greubel, <a href="/A033593/b033593.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: (1 -5*x +115*x^2 +345*x^3 +120*x^4)/(1-x)^5. - _R. J. Mathar_, Jan 30 2011
%F From _G. C. Greubel_, Mar 05 2020: (Start)
%F a(n) = n^4* Pochhammer((n-1)/n, 4).
%F E.g.f.: (1 - x + 53*x^2 + 94*x^3 + 24*x^4)*exp(x). (End)
%F From _Amiram Eldar_, Mar 11 2022: (Start)
%F Sum_{n>=2} 1/a(n) = 29/36 + (4/3 - 3*sqrt(3)/4)*Pi - 12*log(2) + 27*log(3)/4.
%F Sum_{n>=2} (-1)^n/a(n) = (1 + 4*sqrt(2)/3 - 3*sqrt(3)/2)*Pi + 14*log(2)/3 - 4*sqrt(2)*log(2)/3 + 8*sqrt(2)*log(2-sqrt(2))/3 - 29/36. (End)
%p 1, seq( n^4*pochhammer((n-1)/n, 4), n=1..40); # _G. C. Greubel_, Mar 05 2020
%t Table[1-10 n+35 n^2-50 n^3+24 n^4,{n,0,40}] (* or *) LinearRecurrence[{5,-10, 10,-5,1}, {1,0,105,880,3465}, 40] (* _Harvey P. Dale_, Jan 29 2011 & Apr 26 2011 *)
%o (Magma) [ 24*n^4-50*n^3+35*n^2-10*n+1: n in [0..40]]; // _Vincenzo Librandi_, Jan 30 2011
%o (Magma) [&*[s*n-1: s in [1..4]]: n in [0..40]]; // _Bruno Berselli_, May 23 2011
%o (PARI) a(n)=24*n^4-50*n^3+35*n^2-10*n+1 \\ _Charles R Greathouse IV_, May 23 2011
%o (Sage) [1]+[n^4*rising_factorial((n-1)/n, 4) for n in (1..40)] # _G. C. Greubel_, Mar 05 2020
%Y a(n) = A011245(-n).
%Y Cf. A005384, A033594, A067256, A067257.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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