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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

Table of n, a(n) for n=0..39.

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

Cf. A230339/A230328, A300299.

Sequence in context: A057183 A076293 A227276 * A273745 A263264 A072199

Adjacent sequences:  A300295 A300296 A300297 * A300299 A300300 A300301

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Apr 05 2018

STATUS

approved

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Last modified September 20 16:50 EDT 2021. Contains 347586 sequences. (Running on oeis4.)