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A033593
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a(n) = (n-1)*(2*n-1)*(3*n-1)*(4*n-1).
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3
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1, 0, 105, 880, 3465, 9576, 21505, 42120, 74865, 123760, 193401, 288960, 416185, 581400, 791505, 1053976, 1376865, 1768800, 2238985, 2797200, 3453801, 4219720, 5106465, 6126120, 7291345, 8615376, 10112025, 11795680, 13681305, 15784440, 18121201, 20708280, 23562945
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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G.f.: (1 -5*x +115*x^2 +345*x^3 +120*x^4)/(1-x)^5. - R. J. Mathar, Jan 30 2011
a(n) = n^4* Pochhammer((n-1)/n, 4).
E.g.f.: (1 - x + 53*x^2 + 94*x^3 + 24*x^4)*exp(x). (End)
Sum_{n>=2} 1/a(n) = 29/36 + (4/3 - 3*sqrt(3)/4)*Pi - 12*log(2) + 27*log(3)/4.
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)
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MAPLE
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1, seq( n^4*pochhammer((n-1)/n, 4), n=1..40); # G. C. Greubel, Mar 05 2020
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MATHEMATICA
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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 *)
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PROG
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(Magma) [ 24*n^4-50*n^3+35*n^2-10*n+1: n in [0..40]]; // Vincenzo Librandi, Jan 30 2011
(Magma) [&*[s*n-1: s in [1..4]]: n in [0..40]]; // Bruno Berselli, May 23 2011
(Sage) [1]+[n^4*rising_factorial((n-1)/n, 4) for n in (1..40)] # G. C. Greubel, Mar 05 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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