|
| |
|
|
A300298
|
|
Numerators of r(n) := Sum_{k=0..n-1} 1/Product_{j=0..4} (k + j + 1), for n >= 0, with r(0) = 0.
|
|
1
|
|
|
|
0, 1, 7, 17, 23, 125, 209, 329, 247, 119, 125, 341, 1819, 793, 3059, 3875, 1211, 187, 1219, 4427, 10625, 12649, 4983, 17549, 10237, 11875, 6851, 1311, 35959, 40919, 46375, 17453, 7363, 16511, 36907, 41125, 30463, 101269, 111929, 123409
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
The denominators are given in A300299.
The sum given in the name is computed using a telescopic sum. See the general recipe given in the Jolley reference, (201), p. 38.
|
|
|
REFERENCES
|
L. B. W. Jolley, Summation of Series, Dover Publications, 2nd rev. ed., 1961, p. 38, (201).
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = numerator(r(n)), with the result of the sum given in the name r(n) = n*(50 + 35*n + 10*n^2 + n^3)/(96*(1 + n)*(2 + n)*(n + 3)*(4 + n)), n >= 0.
This results from r(n) = 1/96 - 1/(4*(1+n)*(2+n)*(n+3)*(4+n)).
G.f. for rationals {r(n)}_{n >= 0}: (1/96)*(1 - hypergeometric([1, 4], [5], -x/(1-x)))/(1-x)
= (-x*(12 - 42*x + 52*x^2 - 25*x^3) + 12*(1 - x)^4*log(1/(1-x))) / (288*x^4*(1-x)).
|
|
|
EXAMPLE
|
The sum begins: 0 + 1/(1*2*3*4*5) + 1/(2*3*4*5*6) + ... = 0 + 1/120 + 1/720 + 1/2520 + 1/6720 + 1/15120 + 1/30240 + ...
The rationals r(n) (partial sums) begin: 0/1, 1/120, 7/720, 17/1680, 23/2240, 125/12096, 209/20160, 329/31680, 247/23760, 119/11440, 125/12012, 341/32760, ...
|
|
|
MATHEMATICA
|
Table[Numerator[n (50 + 35 n + 10 n^2 + n^3) / (96 (1 + n) (2 + n) (n + 3) (4 + n))], {n, 0, 50}] (* Vincenzo Librandi, Apr 06 2018 *)
|
|
|
PROG
|
(GAP) List(List([0..40], n->Sum([0..n-1], k->1/(Product([0..4], j->k+j+1)))), NumeratorRat); # Muniru A Asiru, Apr 05 2018
(PARI) a(n) = numerator(sum(k=0, n-1, prod(j=0, 4, (k+j+1))^(-1))); \\ Altug Alkan, Apr 05 2018
(Magma) [Numerator(n*(50+35*n+10*n^2+n^3)/(96*(1+n)*(2+n)*(n+3)*(4+n))): n in [0..50]]; // Vincenzo Librandi, Apr 06 2018
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,frac,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
