A002455 Central factorial numbers. (Formerly M5103 N2210)

0, 1, 20, 784, 52480, 5395456, 791691264, 157294854144, 40683662475264, 13288048674471936, 5349739088314368000, 2603081566154391552000, 1506057980251484454912000, 1021944601582419125993472000

Offset

0,3

References

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

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.

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

T. D. Noe, Table of n, a(n) for n=0..50

T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 7.

Index entries for sequences related to factorial numbers

Formula

(-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

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)

a(n) = 4*(2*n^2 - 2*n + 1)*a(n-1) - 16*(n-1)^4*a(n-2). - Vaclav Kotesovec, Feb 23 2015

a(n) ~ Pi^3 * 2^(2*n-2) * n^(2*n+1) / (3 * exp(2*n)). - Vaclav Kotesovec, Feb 23 2015

Example

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

Mathematica

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

Prog

(PARI) a(n)=if(n<0, 0, (2*n+2)!*polcoeff(asin(x+O(x^(2*n+3)))^4/4!, 2*n+2))
(PARI) a(n)=-(-1)^n*polcoeff(prod(k=0, 2*n, x+2*k-2*n), 3)
(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
(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
(GAP) List([0..20], n-> 4^(n-1)*(Factorial(n))^2*Sum([1..n], k-> 1/k^2)) # G. C. Greubel, Jul 04 2019

Crossrefs

Cf. A001819, A001824, A001825, A049033.

Sequence in context: A179712 A012802 A049214 * A322890 A267671 A281777

Adjacent sequences: A002452 A002453 A002454 * A002456 A002457 A002458

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

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

Status

approved