cp's OEIS Frontend

A001825 Central factorial numbers: 2nd subdiagonal of A008956.

%I A001825 M5250 N2284 #32 Jan 13 2025 10:46:38
%S A001825 1,35,1974,172810,21967231,3841278805,886165820604,261042753755556,
%T A001825 95668443268795341,42707926241367380631,22821422608929422854674,
%U A001825 14384681946935352617964750,10562341153570752891930640875
%N A001825 Central factorial numbers: 2nd subdiagonal of A008956.
%D A001825 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A001825 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001825 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001825 T. D. Noe, <a href="/A001825/b001825.txt">Table of n, a(n) for n=0..50</a>
%F A001825 E.g.f.: (arcsin x)^5; that is, a_k is the coefficient of x^(2*k+5) in (arcsin x)^5 multiplied by (2*k+5)! and divided by 5!. - Joe Keane (jgk(AT)jgk.org)
%F A001825 (-1)^(n-2)*a(n-2) is the coefficient of x^4 in prod(k=1, 2*n, x+2*k-2*n-1). - _Benoit Cloitre_ and _Michael Somos_, Nov 22 2002
%F A001825 a(n) = det(V(i+3,j+2), 1 <= i,j <= n), where V(n,k) are central factorial numbers of the second kind with odd indices (A008958). - _Mircea Merca_, Apr 06 2013
%F A001825 a(n) = (12*n^2 + 12*n + 11)*a(n-1) - (4*n^2 + 3)*(12*n^2 + 1)*a(n-2) + (2*n - 1)^6*a(n-3). - _Vaclav Kotesovec_, Feb 23 2015
%F A001825 a(n) ~ Pi^4 * n^(2*n+4) * 2^(2*n-2) / (3*exp(2*n)). - _Vaclav Kotesovec_, Feb 23 2015
%e A001825 (arcsin x)^5 = x^5 + 5/6*x^7 + 47/72*x^9 + 1571/3024*x^11 + ...
%t A001825 Table[(2*n+5)!/5! * SeriesCoefficient[ArcSin[x]^5,{x,0,2*n+5}], {n,0,20}] (* _Vaclav Kotesovec_, Feb 23 2015 *)
%Y A001825 Cf. A001824, A002455, A049033.
%Y A001825 Right-hand column 3 in triangle A008956.
%K A001825 nonn
%O A001825 0,2
%A A001825 _N. J. A. Sloane_
%E A001825 More terms from Joe Keane (jgk(AT)jgk.org)