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A001824 Central factorial numbers.
(Formerly M4749 N2031)
10

%I M4749 N2031 #53 Nov 18 2017 04:44:00

%S 1,10,259,12916,1057221,128816766,21878089479,4940831601000,

%T 1432009163039625,518142759828635250,228929627246078500875,

%U 121292816354463333793500,75908014254880833434338125,55399444912646408707007883750,46636497509226736668824289999375

%N Central factorial numbers.

%D T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 223, Problem 2.

%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Seiichi Manyama, <a href="/A001824/b001824.txt">Table of n, a(n) for n = 0..224</a> (terms 0..50 from T. D. Noe)

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%F 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)

%F a(n) = ((2*n+1)!!)^2 * Sum_{k=0..n} (2*k+1)^(-2).

%F a(n) ~ Pi^2*n^2*2^(2*n)*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002

%F (-1)^(n-1)*a(n-1) is the coefficient of x^2 in Product_{k=1..2*n} (x + 2*k - 2*n - 1). - _Benoit Cloitre_ and _Michael Somos_, Nov 22 2002

%F 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

%F Recurrence: a(n) = 2*(4*n^2+1)*a(n-1) - (2*n-1)^4*a(n-2). - _Vladimir Reshetnikov_, Oct 13 2016

%F Limit_{n->infinity} a(n)/((2n+1)!!)^2 = Pi^2/8. - _Daniel Suteu_, Oct 31 2017

%e (arcsin x)^3 = x^3 + 1/2*x^5 + 37/120*x^7 + 3229/15120*x^9 + ...

%t 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 *)

%t 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 *)

%Y Cf. A002455, A001825, A049033.

%Y Right-hand column 2 in triangle A008956.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_

%E More terms from Joe Keane (jgk(AT)jgk.org)

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)