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Numerators of r(n) := Sum_{k=0..n-1} 1/Product_{j=0..4} (k + j + 1), for n >= 0, with r(0) = 0.
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%I #17 Sep 08 2022 08:46:20

%S 0,1,7,17,23,125,209,329,247,119,125,341,1819,793,3059,3875,1211,187,

%T 1219,4427,10625,12649,4983,17549,10237,11875,6851,1311,35959,40919,

%U 46375,17453,7363,16511,36907,41125,30463,101269,111929,123409

%N Numerators of r(n) := Sum_{k=0..n-1} 1/Product_{j=0..4} (k + j + 1), for n >= 0, with r(0) = 0.

%C The denominators are given in A300299.

%C The sum given in the name is computed using a telescopic sum. See the general recipe given in the Jolley reference, (201), p. 38.

%D L. B. W. Jolley, Summation of Series, Dover Publications, 2nd rev. ed., 1961, p. 38, (201).

%F 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.

%F This results from r(n) = 1/96 - 1/(4*(1+n)*(2+n)*(n+3)*(4+n)).

%F G.f. for rationals {r(n)}_{n >= 0}: (1/96)*(1 - hypergeometric([1, 4], [5], -x/(1-x)))/(1-x)

%F = (-x*(12 - 42*x + 52*x^2 - 25*x^3) + 12*(1 - x)^4*log(1/(1-x))) / (288*x^4*(1-x)).

%e 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 + ...

%e 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, ...

%t 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 *)

%o (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

%o (PARI) a(n) = numerator(sum(k=0, n-1, prod(j=0, 4, (k+j+1))^(-1))); \\ _Altug Alkan_, Apr 05 2018

%o (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

%Y Cf. A230339/A230328, A300299.

%K nonn,frac,easy

%O 0,3

%A _Wolfdieter Lang_, Apr 05 2018