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A002455 Central factorial numbers.
(Formerly M5103 N2210)
7
0, 1, 20, 784, 52480, 5395456, 791691264, 157294854144, 40683662475264, 13288048674471936, 5349739088314368000, 2603081566154391552000, 1506057980251484454912000, 1021944601582419125993472000 (list; graph; refs; listen; history; text; 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). - 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-Fran├žois 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 * A267671 A281777 A041763

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

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Last modified December 14 12:28 EST 2018. Contains 318097 sequences. (Running on oeis4.)