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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). 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. 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 EXTENSIONS More terms from Joe Keane (jgk(AT)jgk.org) STATUS approved

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Last modified October 23 20:32 EDT 2019. Contains 328373 sequences. (Running on oeis4.)