%I M5103 N2210 #34 Sep 08 2022 08:44:30

%S 0,1,20,784,52480,5395456,791691264,157294854144,40683662475264,

%T 13288048674471936,5349739088314368000,2603081566154391552000,

%U 1506057980251484454912000,1021944601582419125993472000

%N Central factorial numbers.

%D B. Berndt, Ramanujan's Notebooks, Part I, page 263.

%D A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 110.

%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 T. D. Noe, <a href="/A002455/b002455.txt">Table of n, a(n) for n=0..50</a>

%H T. R. Van Oppolzer, <a href="http://www.archive.org/stream/lehrbuchzurbahnb02oppo#page/7/mode/1up">Lehrbuch zur Bahnbestimmung der Kometen und Planeten</a>, Vol. 2, Engelmann, Leipzig, 1880, p. 7.

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

%F (-1)^(n-1)*a(n) is the coefficient of x^3 in Product_{k=0..2*n} (x+2*k-2*n). - _Benoit Cloitre_ and _Michael Somos_, Nov 22 2002

%F E.g.f.: (arcsin x)^4; that is, a_k is the coefficient of x^(2*k+2) in (arcsin x)^4 multiplied by (2*k+2)! and divided by 4! Also a(n) = 2^(2*n-2)*(n!)^2 * Sum_{k=1..n} 1/k^2. - Joe Keane (jgk(AT)jgk.org)

%F a(n) = 4*(2*n^2 - 2*n + 1)*a(n-1) - 16*(n-1)^4*a(n-2). - _Vaclav Kotesovec_, Feb 23 2015

%F a(n) ~ Pi^3 * 2^(2*n-2) * n^(2*n+1) / (3 * exp(2*n)). - _Vaclav Kotesovec_, Feb 23 2015

%e (arcsin x)^4 = x^4 + 2/3*x^6 + 7/15*x^8 + 328/945*x^10 + ...

%t nmax = 13; coes = CoefficientList[ Series[ ArcSin[x]^4, {x, 0, 2*nmax + 2}], x]* Range[0, 2*nmax + 2]!/24; a[n_] := coes[[2*n + 3]]; Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Dec 08 2011 *)

%t Table[4^(n-1)*(n!)^2*HarmonicNumber[n,2], {n,0,20}] (* _G. C. Greubel_, Jul 04 2019 *)

%o (PARI) a(n)=if(n<0,0,(2*n+2)!*polcoeff(asin(x+O(x^(2*n+3)))^4/4!,2*n+2))

%o (PARI) a(n)=-(-1)^n*polcoeff(prod(k=0,2*n,x+2*k-2*n),3)

%o (Magma) [0] cat [4^(n-1)*(Factorial(n))^2*(&+[1/k^2: k in [1..n]]): n in [1..20]]; // _G. C. Greubel_, Jul 04 2019

%o (Sage) [4^(n-1)*(factorial(n))^2*sum(1/k^2 for k in (1..n)) for n in (0..20)] # _G. C. Greubel_, Jul 04 2019

%o (GAP) List([0..20], n-> 4^(n-1)*(Factorial(n))^2*Sum([1..n], k-> 1/k^2)) # _G. C. Greubel_, Jul 04 2019

%Y Cf. A001819, A001824, A001825, A049033.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_

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