|
| |
|
|
A002455
|
|
Central factorial numbers.
(Formerly M5103 N2210)
|
|
5
| |
|
|
0, 1, 20, 784, 52480, 5395456, 791691264, 157294854144, 40683662475264, 13288048674471936, 5349739088314368000, 2603081566154391552000, 1506057980251484454912000, 1021944601582419125993472000
(list; graph; refs; listen; history; internal format)
|
|
|
|
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).
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 7.
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..50
Index entries for sequences related to factorial numbers
|
|
|
FORMULA
| (-1)^(n-1)*a(n) is the coefficient of x^3 in prod(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 ] k^(-2) - from Joe Keane (jgk(AT)jgk.org)
|
|
|
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}] (* From Jean-François Alcover, Dec 08 2011 *)
|
|
|
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)
|
|
|
CROSSREFS
| Cf. A001819, A001824, A001825, A049033.
Sequence in context: A179712 A012802 A049214 * A041763 A041760 A117798
Adjacent sequences: A002452 A002453 A002454 * A002456 A002457 A002458
|
|
|
KEYWORD
| nonn,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| More terms from Joe Keane (jgk(AT)jgk.org)
|
| |
|
|
