# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a002455 Showing 1-1 of 1 %I A002455 M5103 N2210 #34 Sep 08 2022 08:44:30 %S A002455 0,1,20,784,52480,5395456,791691264,157294854144,40683662475264, %T A002455 13288048674471936,5349739088314368000,2603081566154391552000, %U A002455 1506057980251484454912000,1021944601582419125993472000 %N A002455 Central factorial numbers. %D A002455 B. Berndt, Ramanujan's Notebooks, Part I, page 263. %D A002455 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 A002455 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217. %D A002455 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A002455 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A002455 T. D. Noe, Table of n, a(n) for n=0..50 %H A002455 T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 7. %H A002455 Index entries for sequences related to factorial numbers %F A002455 (-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 A002455 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 A002455 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 A002455 a(n) ~ Pi^3 * 2^(2*n-2) * n^(2*n+1) / (3 * exp(2*n)). - _Vaclav Kotesovec_, Feb 23 2015 %e A002455 (arcsin x)^4 = x^4 + 2/3*x^6 + 7/15*x^8 + 328/945*x^10 + ... %t A002455 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 A002455 Table[4^(n-1)*(n!)^2*HarmonicNumber[n,2], {n,0,20}] (* _G. C. Greubel_, Jul 04 2019 *) %o A002455 (PARI) a(n)=if(n<0,0,(2*n+2)!*polcoeff(asin(x+O(x^(2*n+3)))^4/4!,2*n+2)) %o A002455 (PARI) a(n)=-(-1)^n*polcoeff(prod(k=0,2*n,x+2*k-2*n),3) %o A002455 (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 A002455 (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 A002455 (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 A002455 Cf. A001819, A001824, A001825, A049033. %K A002455 nonn,easy,nice %O A002455 0,3 %A A002455 _N. J. A. Sloane_ %E A002455 More terms from Joe Keane (jgk(AT)jgk.org) # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE