A001825 Central factorial numbers: 2nd subdiagonal of A008956.

1, 35, 1974, 172810, 21967231, 3841278805, 886165820604, 261042753755556, 95668443268795341, 42707926241367380631, 22821422608929422854674, 14384681946935352617964750, 10562341153570752891930640875

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			(arcsin x)^5 = x^5 + 5/6*x^7 + 47/72*x^9 + 1571/3024*x^11 + ...
		

References

• 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

Crossrefs

Cf. A001824, A002455, A049033.

Right-hand column 3 in triangle A008956.

Programs

Mathematica

Table[(2*n+5)!/5! * SeriesCoefficient[ArcSin[x]^5,{x,0,2*n+5}], {n,0,20}] (* Vaclav Kotesovec, Feb 23 2015 *)

Formula

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)

(-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

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

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

a(n) ~ Pi^4 * n^(2*n+4) * 2^(2*n-2) / (3*exp(2*n)). - Vaclav Kotesovec, Feb 23 2015

Extensions

More terms from Joe Keane (jgk(AT)jgk.org)