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 A303108 a(n) = (2*n-1)*a(n-1) - (n-2)!, with a(1) = 2, n > 1. 2
 2, 5, 24, 166, 1488, 16344, 212352, 3184560, 54132480, 1028476800, 21597649920, 496742319360, 12418518067200, 335299508812800, 9723679528550400, 301433978206771200, 9947319973149081600, 348156178137427968000, 12881778235397406720000, 502389344778125156352000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Chai Wah Wu, Table of n, a(n) for n = 1..404 Travis Sherman, Summation of Glaisher- and Apery-like Series, University of Arizona, May 23 2000, p. 15, (3.94) - (3.98). FORMULA a(n) = f1(n)*2*(n-1)!, where f1(n) corresponds to the x values such that Sum_{k>=0} 2^k/(binomial(2*k,k)*(k+n))) = x*Pi - y*Pi^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) = (1/2^(n+2))*((2*n-1)!/((n-1)!)^2). a(n) = 3*(n-1)*a(n-1) - (2n-3)*(n-2)*a(n-2) for n > 2. - Chai Wah Wu, Apr 20 2018 EXAMPLE Examples ((3.94) - (3.98)) at page 15 in Links section as follows, respectively. For n=1, f1(1) = 1, so a(1) = 2. For n=2, f1(2) = 5/2, so a(2) = 5. For n=3, f1(3) = 6, so a(3) = 24. For n=4, f1(4) = 83/6, so a(4) = 166. For n=5, f1(5) = 31, so a(5) = 1488. MATHEMATICA RecurrenceTable[{a[n] == (2*n-1)*a[n-1] - (n-2)!, a[1] == 2}, a, {n, 1, 15}] (* Altug Alkan, Apr 20 2018 *) nmax = 15; Table[CoefficientList[Expand[FunctionExpand[Table[Sum[ 2^j/(Binomial[2*j, j]*(j + m)), {j, 0, Infinity}], {m, 1, nmax}]]], Pi][[n, 2]]*2*(n-1)!, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 20 2018 *) PROG (PARI) a=vector(20); a[1]=2; for(n=2, #a, a[n]=(2*n-1)*a[n-1] - (n-2)!); a (Python) A303108_list = [2, 5] for n in range(3, 501):     A303108_list.append(3*(n-1)*A303108_list[-1]-(2*n-3)*(n-2)*A303108_list[-2]) # Chai Wah Wu, Apr 20 2018 (MAGMA) I:=[2, 5]; [n le 2 select I[n] else 3*(n-1)*Self(n-1)- (2*n-3)*(n-2)*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Apr 20 2018 CROSSREFS Cf. A303109. Sequence in context: A086635 A020124 A012249 * A052808 A137157 A025134 Adjacent sequences:  A303105 A303106 A303107 * A303109 A303110 A303111 KEYWORD nonn AUTHOR Detlef Meya, Apr 18 2018 STATUS approved

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Last modified May 20 12:33 EDT 2019. Contains 323422 sequences. (Running on oeis4.)