cp's OEIS Frontend

A001824 Central factorial numbers: 1st subdiagonal of A008956.

%I A001824 M4749 N2031 #55 Jan 13 2025 10:23:10
%S A001824 1,10,259,12916,1057221,128816766,21878089479,4940831601000,
%T A001824 1432009163039625,518142759828635250,228929627246078500875,
%U A001824 121292816354463333793500,75908014254880833434338125,55399444912646408707007883750,46636497509226736668824289999375
%N A001824 Central factorial numbers: 1st subdiagonal of A008956.
%D A001824 T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 223, Problem 2.
%D A001824 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A001824 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001824 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001824 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 A001824 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A001824 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 A001824 a(n) = ((2*n+1)!!)^2 * Sum_{k=0..n} (2*k+1)^(-2).
%F A001824 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 A001824 (-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 A001824 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 A001824 Recurrence: a(n) = 2*(4*n^2+1)*a(n-1) - (2*n-1)^4*a(n-2). - _Vladimir Reshetnikov_, Oct 13 2016
%F A001824 Limit_{n->infinity} a(n)/((2n+1)!!)^2 = Pi^2/8. - _Daniel Suteu_, Oct 31 2017
%e A001824 (arcsin x)^3 = x^3 + 1/2*x^5 + 37/120*x^7 + 3229/15120*x^9 + ...
%t A001824 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 A001824 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 A001824 Cf. A002455, A001825, A049033.
%Y A001824 Right-hand column 2 in triangle A008956.
%K A001824 nonn,easy,nice
%O A001824 0,2
%A A001824 _N. J. A. Sloane_
%E A001824 More terms from Joe Keane (jgk(AT)jgk.org)