A118431 -id:A118431 - cp's OEIS frontend

A118432 Denominator of sum of reciprocals of first n 5-simplex numbers A000389.

1, 6, 14, 56, 504, 168, 264, 198, 286, 1001, 273, 1456, 1904, 2448, 15504, 969, 1197, 2926, 3542, 42504, 10120, 11960, 14040, 8190, 47502, 5481, 6293, 28768, 32736, 185504, 41888, 11781, 13209, 29526, 164502, 73112, 81016

Offset: 1

Jonathan Vos Post, Apr 28 2006

Numerators are A118431. Fractions are: 1/1, 7/6, 17/14, 69/56, 625/504, 209/168, 329/264, 247/198, 357/286, 1250/1001, 341/273, 1819/1456, 2379/1904, 3059/2448, 19375/15504, 1211/969, 1496/1197, 3657/2926, 4427/3542, 53125/42504, 12649/10120, 14949/11960, 17549/14040, 10237/8190, 59375/47502, 6851/5481, 7866/6293, 35959/28768, 40919/32736, 231875/185504, 52359/41888, 14726/11781, 16511/13209, 36907/29526, 205625/164502, 91389/73112, 101269/81016. The denominator of sum of reciprocals of first n triangular numbers is A026741. The denominator of sum of reciprocals of first n tetrahedral numbers is A118392. The denominator of sum of reciprocals of first n pentatope numbers is A118412.

			a(1) = 1 = denominator of 1/1.
a(2) = 6 = denominator of 7/6 = 1/1 + 1/6.
a(3) = 14 = denominator of 17/14 = 1/1 + 1/6 + 1/21.
a(4) = 56 = denominator of 69/56 = 1/1 + 1/6 + 1/21 + 1/56.
a(5) = 42 = denominator of 55/42 = 1/1 + 1/6 + 1/21 + 1/56 + 1/126.
a(10) = 1001 = denominator of 1250/1001 = 1/1+ 1/6 + 1/21 + 1/56 + 1/126 + 1/252 + 1/462 + 1/792 + 1/1287 + 1/2002.
a(20) = 42504 = denominator of 53125/42504 = 1/1 + 1/6 + 1/21 + 1/56 + 1/126 + 1/252 + 1/462 + 1/792 + 1/1287 + 1/2002 + 1/3003 + 1/4368 + 1/6188 + 1/8568 + 1/11628 + 1/15504 + 1/20349 + 1/26334 + 1/33649 + 1/42504.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000332, A000389, A022998, A026741, A118391, A118391, A118411, A118412, A118431.

Denominator[Accumulate[1/Binomial[Range[5,50],5]]] (* Harvey P. Dale, Jul 17 2016 *)

A118411(n)/A118412(n) = Sum_{i=1..n} (1/A000389(n)).
A118411(n)/A118412(n) = Sum_{i=1..n} (1/C(n,5)).
A118411(n)/A118412(n) = Sum_{i=1..n} (1/(n*(n+1)*(n+2)*(n+3)*(n+4)/120)).

A118583 Numerator of sum of first p reciprocals of p-simplex numbers divided by p^4, where p = prime(n) for n > 2.

Original entry on oeis.org

1, 5, 53, 789, 237493, 2576561, 338350897, 616410400171, 2603853251291, 5745400286707685, 3081677433937346539, 41741941495866750557, 7829195555633964779233, 21066131970056662377432067

Offset: 3

Alexander Adamchuk, May 09 2007

nonn

			Prime(3) = 5.
a(3) = 1 because A118431(5)/5^4 = 1, where A118431(5) = Numerator[ 1/C(4+1,5) + 1/C(4+2,5) + 1/C(4+3,5) + 1/C(4+4,5) +1/C(4+5,5) ] = Numerator[ 1/1 + 1/6 + 1/21 + 1/56 + 1/126 ] = 625.

G. C. Greubel, Table of n, a(n) for n = 3..250
Eric Weisstein's World of Mathematics, Composition.
Eric Weisstein's World of Mathematics, Tetrahedral Number.
Eric Weisstein's World of Mathematics, Triangular Number.

Cf. A022998 = Numerator of sum of reciprocals of first n triangular numbers
Cf. A118391 = Numerator of sum of reciprocals of first n tetrahedral numbers A000292.
Cf. A118431 = Numerator of sum of reciprocals of first n 5-simplex numbers A000389.

Table[Numerator[Sum[1 /Binomial[ n+Prime[k]-1, Prime[k]], {n,1,Prime[k]} ]]/ Prime[k]^4, {k,3,25}]

for(n=3,10, print1(numerator(sum(k=1,prime(n), 1/(binomial(k+ prime(n)-1, prime(n)))))/prime(n)^4, ", ")) \\ G. C. Greubel, Nov 25 2017

a(n) = numerator( Sum_{k=1..prime(n)} ( 1/binomial( k + prime(n) - 1, prime(n) ) ))/prime(n)^4 for n > 2.

cp's OEIS Frontend

A118432 Denominator of sum of reciprocals of first n 5-simplex numbers A000389.

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