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 A257583 a(0)=4; thereafter a(n)=8*n*(2*n-1)*a(n-1). 1
 4, 32, 1536, 184320, 41287680, 14863564800, 7847962214400, 5713316492083200, 5484783832399872000, 6713375410857443328000, 10204330624503313858560000, 18857602994082124010618880000, 41637587410933329815446487040000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS It appears that for n>=1, Sum_{k >= 0} ( 1/(4k+1)^(2n+1) - 1/(4k+3)^(2n+1) ) = E_n*Pi^(2n+1)/a(n), where E_n = A000364(n) are the Euler numbers. For n=0, one must subtract 2/3 from the right-hand side. It appears that the entries are the numerators of the 2n-th coefficients of the Taylor series expansion of f[x] = x Log[x]+(1-x)Log[1-x] about x=1/2, beginning at n=1, odd derivatives being zero, denominators being 2n!. In other words, the 2n-order derivative of f[x] evaluated at x=1/2 for n=1,2,3... (See Mathematica below). - Paul Reiser, May 23 2015 LINKS Jean-Claude Babois, Personal communication to N. J. A. Sloane, Apr 29 2015 (Pages 4 and 1, Pages 2 and 3) FORMULA The ratio of successive terms is 8*A000384(n). a(n) = 2^(2*n+2)*(2*n)!. - Vaclav Kotesovec, May 05 2015 EXAMPLE For n=0, Sum_{k >= 0} ( 1/(4k+1)^(2n+1) - 1/(4k+3)^(2n+1) ) = 2*sum(1/((4*n+1)*(4*n+3)),n=1..infinity) = Pi/4 - 2/3 = E_0*Pi/a(0) - 2/3. For n=1, the sum is Pi^3/32 = E_1*Pi^3/a(1). For n=2, the sum is 5*Pi^5/1536 = E_2*Pi^5/a(2). MAPLE f:=proc(n) option remember; if n=0 then 4 else 8*n*(2*n-1)*f(n-1); fi; end; [seq(f(n), n=0..20)]; MATHEMATICA lst={4}; Do[AppendTo[lst, 8*n*(2*n-1)*Last[lst]], {n, 1, 12}]; lst (* Ivan N. Ianakiev, May 04 2015 *) Table[2^(2*n+2)*(2*n)!, {n, 0, 15}] (* Vaclav Kotesovec, May 05 2015 *) f[x_] := x Log[x] + (1 - x) Log[1 - x]; Table[D[f[x], {x, 2n}] /. x -> 1/2, {n, 1, 14}](* Paul Reiser, May 23 2015*) PROG (MAGMA) [2^(2*n+2)*Factorial(2*n): n in [0..15]]; // Vincenzo Librandi, May 23 2015 CROSSREFS Cf. A000364, A000384. Sequence in context: A028369 A081790 A053005 * A258122 A012092 A027639 Adjacent sequences:  A257580 A257581 A257582 * A257584 A257585 A257586 KEYWORD nonn AUTHOR N. J. A. Sloane, May 04 2015 STATUS approved

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Last modified June 6 17:28 EDT 2020. Contains 334830 sequences. (Running on oeis4.)