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A001824 Central factorial numbers.
(Formerly M4749 N2031)
9
1, 10, 259, 12916, 1057221, 128816766, 21878089479, 4940831601000, 1432009163039625, 518142759828635250, 228929627246078500875, 121292816354463333793500, 75908014254880833434338125, 55399444912646408707007883750, 46636497509226736668824289999375 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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

EXAMPLE

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

MATHEMATICA

a[n_] = (2n+1)!!^2 (Pi^2 - 2 PolyGamma[1, n+3/2])/8; a /@ Range[0, 12] // Simplify (* Jean-Fran├žois 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 *)

CROSSREFS

Cf. A002455, A001825, A049033.

Right-hand column 2 in triangle A008956.

Sequence in context: A126468 A024293 A120268 * A024294 A183406 A084999

Adjacent sequences:  A001821 A001822 A001823 * A001825 A001826 A001827

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified November 12 07:00 EST 2019. Contains 329052 sequences. (Running on oeis4.)