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 A302866 a(n) = 2*(n*(2*n+1)*b(n-1) + (-1)^(n-1)*(2*n+1)*((n-1)!)^2), with a(0) = 0, n > 1. 1
 0, 6, 110, 4676, 336024, 36975312, 5767774272, 1211248149120, 329458632906240, 112674914230625280, 47323458446223052800, 23945670579525578956800, 14367402268047801262080000, 10085916404559453257318400000, 8189764118253280338411847680000, 7616480630446754091029242183680000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Travis Sherman, Summation of Glaisher- and Apery-like Series, University of Arizona, May 23 2000, p. 14, (3.84) - (3.88). FORMULA a(n-1) = f3(n)*(-1)^n*((n-1)!)^2, where f3(n) corresponds to the z values such that Sum_{k>=0} (-1)^k/(binomial(2*k,k)*(k+n)) = x*sqrt(5)*log(phi) + y*(log(phi))^2 + z. (See examples for connection with a(n) in terms of material at Links section.) f2(n) corresponds to the y values, so f2(n) = (-1)^n*4*((2*n-1)!/((n-1)!)^2). EXAMPLE Examples ((3.83) - (3.88)) at page 14 in Links section as follows, respectively. For n=1, f3(1) = 0, so a(0) = 0. For n=2, f3(2) = 6, so a(1) = 6. For n=3, f3(3) = -55/2, so a(2) = 110. For n=4, f3(4) = 1169/9, so a(3) = 4676. For n=5, f3(5) = -4667/8, so a(4) = 336024. MATHEMATICA Fold[Append[#1, 2 (#2 (2 #2 + 1) #1[[#2]] + (-1)^(#2 - 1) (2 #2 + 1) ((#2 - 1)!)^2)] &, {0}, Range@ 19] (* Michael De Vlieger, Apr 14 2018 *) nmax = 15; Table[CoefficientList[TrigToExp[Expand[FunctionExpand[ Table[Sum[(-1)^j/(Binomial[2*j, j]*(j + m)), {j, 0, Infinity}], {m, 1, nmax}]]]], Log[1/2 + Sqrt[5]/2]][[n, 1]]*(-1)^n*(n - 1)!^2, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 15 2018 *) PROG (PARI) a=vector(20); a[1]=0; for(n=2, #a, a[n]=2*((n-1)*(2*n-1)*a[n-1] + (-1)^(n)*(2*n-1)*((n-2)!)^2)); a \\ CROSSREFS Cf. A302865. Sequence in context: A197325 A197978 A241702 * A243511 A199222 A260026 Adjacent sequences:  A302863 A302864 A302865 * A302867 A302868 A302869 KEYWORD nonn AUTHOR Detlef Meya, Apr 14 2018 STATUS approved

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Last modified July 23 12:19 EDT 2021. Contains 346259 sequences. (Running on oeis4.)