A001824 Central factorial numbers: 1st subdiagonal of A008956.

1, 10, 259, 12916, 1057221, 128816766, 21878089479, 4940831601000, 1432009163039625, 518142759828635250, 228929627246078500875, 121292816354463333793500, 75908014254880833434338125, 55399444912646408707007883750, 46636497509226736668824289999375

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Examples

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

References

• 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).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..224 (terms 0..50 from T. D. Noe)
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A002455, A001825, A049033.

Right-hand column 2 in triangle A008956.

Programs

Mathematica

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 *)

Formula

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

(-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

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

Recurrence: a(n) = 2*(4*n^2+1)*a(n-1) - (2*n-1)^4*a(n-2). - Vladimir Reshetnikov, Oct 13 2016

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

Extensions

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