A033593 a(n) = (n-1)*(2*n-1)*(3*n-1)*(4*n-1).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

a(n) = A011245(-n).

Cf. A005384, A033594, A067256, A067257.

Programs

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

Maple

1, seq( n^4*pochhammer((n-1)/n, 4), n=1..40); # G. C. Greubel, Mar 05 2020

Mathematica

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 *)

PARI

a(n)=24*n^4-50*n^3+35*n^2-10*n+1 \\ Charles R Greathouse IV, May 23 2011

Sage

[1]+[n^4*rising_factorial((n-1)/n, 4) for n in (1..40)] # G. C. Greubel, Mar 05 2020

Formula

G.f.: (1 -5*x +115*x^2 +345*x^3 +120*x^4)/(1-x)^5. - R. J. Mathar, Jan 30 2011

From G. C. Greubel, Mar 05 2020: (Start)

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)

From Amiram Eldar, Mar 11 2022: (Start)

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)