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A118391
Numerator of sum of reciprocals of first n tetrahedral numbers A000292.
7
1, 5, 27, 7, 10, 81, 35, 22, 81, 65, 77, 135, 52, 119, 405, 76, 85, 567, 209, 115, 378, 275, 299, 486, 175, 377, 1215, 217, 232, 1485, 527, 280, 891, 629, 665, 1053, 370, 779, 2457, 430, 451, 2835, 989, 517, 1620, 1127, 1175, 1836, 637, 1325, 4131, 715, 742
OFFSET
1,2
COMMENTS
Denominators are A118392. Fractions are: 1/1, 5/4, 27/20, 7/5, 10/7, 81/56, 35/24, 22/15, 81/55, 65/44, 77/52, 135/91, 52/35, 119/80, 405/272, 76/51, 85/57, 567/380, 209/140, 115/77, 378/253, 275/184, 299/200, 486/325, 175/117, 377/252, 1215/812, 217/145, 232/155, 1485/992.
2n+3 divides a(2n). 2n-1 divides a(2n-1). p divides a(p) for prime p>2. The only primes in a(n) are a(2) = 5 and a(4) = 7. - Alexander Adamchuk, May 08 2007
LINKS
FORMULA
A118391(n)/A118392(n) = Sum_{i=1..n} 1/A000292(n).
A118391(n)/A118392(n) = Sum_{i=1..n} 1/C(n+2,3).
A118391(n)/A118392(n) = Sum_{i=1..n} 6/(n*(n+1)*(n+2)).
a(n) = Numerator( 3*n*(n+3)/(2*(n+1)*(n+2)) ). - Alexander Adamchuk, May 08 2007
EXAMPLE
a(1) = 1 = numerator of 1/1.
a(2) = 5 = numerator of 5/4 = 1/1 + 1/4.
a(3) = 27 = numerator of 27/20 = 1/1 + 1/4 + 1/10.
a(4) = 7 = numerator of 7/5 = 1/1 + 1/4 + 1/10 + 1/20.
a(5) = 10 = numerator of 10/7 = 1/1 + 1/4 + 1/10 + 1/20 + 1/35.
a(20) = 115 = numerator of 115/77 = 1/1 + 1/4 + 1/10 + 1/20 + 1/35 + 1/56 + 1/84 + 1/120 + 1/165 + 1/220 + 1/286 + 1/364 + 1/455 + 1/560 + 1/680 + 1/816 + 1/969 + 1/1140 + 1/1330 + 1/1540.
MAPLE
A118391:= n-> numer(3*n*(n+3)/(2*(n+1)*(n+2))); seq(A118391(n), n=1..60) # G. C. Greubel, Feb 18 2021
MATHEMATICA
Table[ Numerator[3n(n+3)/(2(n+1)(n+2))], {n, 1, 100} ] (* Alexander Adamchuk, May 08 2007 *)
Accumulate[1/Binomial[Range[60]+2, 3]]//Numerator (* Harvey P. Dale, Aug 31 2023 *)
PROG
(PARI) s=0; for(i=3, 50, s+=1/binomial(i, 3); print(numerator(s))) /* Phil Carmody, Mar 27 2012 */
(Sage) [numerator(3*n*(n+3)/(2*(n+1)*(n+2))) for n in (1..60)] # G. C. Greubel, Feb 18 2021
(Magma) [Numerator(3*n*(n+3)/(2*(n+1)*(n+2))): n in [1..60]]; // G. C. Greubel, Feb 18 2021
CROSSREFS
KEYWORD
easy,frac,nonn
AUTHOR
Jonathan Vos Post, Apr 27 2006
EXTENSIONS
More terms from Alexander Adamchuk, May 08 2007
STATUS
approved