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A001824
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Central factorial numbers.
(Formerly M4749 N2031)
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10
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1, 10, 259, 12916, 1057221, 128816766, 21878089479, 4940831601000, 1432009163039625, 518142759828635250, 228929627246078500875, 121292816354463333793500, 75908014254880833434338125, 55399444912646408707007883750, 46636497509226736668824289999375
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OFFSET
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0,2
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REFERENCES
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T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 223, Problem 2.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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E.g.f.: (arcsin x)^3; that is, a_k is the coefficient of x^(2*k+3) in (arcsin x)^3 multiplied by (2*k+3)! and divided by 6. - Joe Keane (jgk(AT)jgk.org)
a(n) = ((2*n+1)!!)^2 * Sum_{k=0..n} (2*k+1)^(-2).
a(n) ~ Pi^2*n^2*2^(2*n)*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
a(n) = det(V(i+2,j+1), 1 <= i,j <= n), where V(n,k) are central factorial numbers of the second kind with odd indices (A008958). - Mircea Merca, Apr 06 2013
Limit_{n->infinity} a(n)/((2n+1)!!)^2 = Pi^2/8. - Daniel Suteu, Oct 31 2017
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EXAMPLE
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(arcsin x)^3 = x^3 + 1/2*x^5 + 37/120*x^7 + 3229/15120*x^9 + ...
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MATHEMATICA
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a[n_] = (2n+1)!!^2 (Pi^2 - 2 PolyGamma[1, n+3/2])/8; a /@ Range[0, 12] // Simplify (* Jean-François Alcover, Apr 22 2011, after Joe Keane *)
With[{nn=30}, Take[(CoefficientList[Series[ArcSin[x]^3, {x, 0, nn}], x] Range[0, nn-1]!)/6, {4, -1, 2}]] (* Harvey P. Dale, Feb 05 2012 *)
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CROSSREFS
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Right-hand column 2 in triangle A008956.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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More terms from Joe Keane (jgk(AT)jgk.org)
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STATUS
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approved
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