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 A302944 a(n) = 4*((2*n-1)*a(n-1) + (-1)^n*(n-2)!), with a(1) = 8, n > 1. 1
 8, 100, 1996, 55896, 2012232, 88538304, 4603991328, 276239482560, 18784284793920, 1427605644499200, 119918874136481280, 11032536420570792960, 1103253642056919628800, 119151393342149235916800, 13821561627689286458265600, 1713873641833471869538099200, 226231320722018281548331622400 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Travis Sherman, Summation of Glaisher- and Apery-like Series, University of Arizona, May 23 2000, p. 14, (3.89) - (3.93). FORMULA a(n) = f1(n)*(-1)^(n-1)*3*(n-1)!, where f1(n) corresponds to the x values such that Sum_{k>=0} (-1)^k/(binomial(2*k,k)*2^k*(k+n)) = x*log(2) + y*(log(2))^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*2^n*((2*n-1)!/((n-1)!)^2). EXAMPLE Examples ((3.89) - (3.93)) at page 14 in Links section as follows, respectively. For n=1, f1(1) = 8/3, so a(1) = 8. For n=2, f1(2) = -100/3, so a(2) = 100. For n=3, f1(3) = 998/3, so a(3) = 1996. For n=4, f1(4) = -9316/3, so a(4) = 55896. For n=5, f1(5) = 83843/3, so a(5) = 2012232. MATHEMATICA RecurrenceTable[{a[n] == 4*((2*n-1)*a[n-1] + (-1)^n*(n-2)!), a[1] == 8}, a, {n, 1, 20}] (* Altug Alkan, Apr 16 2018 *) nmax = 15; Table[CoefficientList[TrigToExp[Expand[FunctionExpand[Table[Sum[(-1)^j/(Binomial[2*j, j]*2^j*(j + m)), {j, 0, Infinity}], {m, 1, nmax}]]]], Log[2]][[n, 2]]*(-1)^(n - 1)*3*(n - 1)!, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 18 2018 *) PROG (PARI) a=vector(20); a[1]=8; for(n=2, #a, a[n]=4*((2*n-1)*a[n-1] + (-1)^n*((n-2)!))); a \\ CROSSREFS Cf. A302945. Sequence in context: A251686 A306032 A274844 * A060570 A215875 A317598 Adjacent sequences:  A302941 A302942 A302943 * A302945 A302946 A302947 KEYWORD nonn AUTHOR Detlef Meya, Apr 16 2018 STATUS approved

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Last modified March 29 02:19 EDT 2020. Contains 333104 sequences. (Running on oeis4.)