A006228 Expansion of e.g.f. exp(arcsin(x)).

1, 1, 1, 2, 5, 20, 85, 520, 3145, 26000, 204425, 2132000, 20646925, 260104000, 2993804125, 44217680000, 589779412625, 9993195680000, 151573309044625, 2898026747200000, 49261325439503125, 1049085682486400000, 19753791501240753125, 463695871658988800000

Offset: 0

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

• nonn

References

• L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 150.
• 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..100
• H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
• Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Crossrefs

Bisections are expansions of sin(arcsinh(x)) and cos(arcsinh(x)).

Bisections are A101927 and A101928.

Row sums of A385343.

Cf. A002019.

Cf. A166741, A166748. - Jaume Oliver Lafont, Oct 24 2009

Programs

Maple

a:= n-> n!*coeff(series(exp(arcsin(x)), x, n+1), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 17 2018

Mathematica

Distribute[ CoefficientList[ Series[ E^ArcSin[x], {x, 0, 21}], x] * Table[ n!, {n, 0, 21}]] (* Robert G. Wilson v, Feb 10 2004 *)
With[{nn=30},CoefficientList[Series[Exp[ArcSin[x]],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Feb 26 2013 *)
Table[FullSimplify[2^(n-2) * (Exp[Pi/2]-(-1)^n*Exp[-Pi/2]) * Gamma[(n-I)/2] * Gamma[(n+I)/2] / Pi], {n, 0, 20}] (* Vaclav Kotesovec, Nov 06 2014 *)

Maxima

a(n):=(n-1)!*sum((if n=m then 1 else if oddp(n-m) then 0 else sum((-1)^k*(sum(binomial(k,j)*2^(1-j)*sum((-1)^((n-m)/2-i)*binomial(j,i)*(j-2*i)^(n-m+j)/(n-m+j)!,i,0,floor(j/2))*(-1)^(k-j),j,1,k))*binomial(k+n-1,n-1),k,1,n-m))/(m-1)!,m,1,n); /* Vladimir Kruchinin, Sep 12 2010 */

Formula

i even: a_i = Product_{j=1..i/2-1} 1 + 4j^2, i odd: a_i = Product_{j=1..(i-1)/2} 2 + 4j(j-1). - Cris Moore (moore(AT)santafe.edu), Jan 31 2001

a(0)=1, a(1)=1, a(n) = (1+(n-2)^2)*a(n-2) for n >= 2. Jaume Oliver Lafont, Oct 24 2009

a(n) = (n-1)!*sum((if n=m then 1 else if oddp(n-m) then 0 else sum((-1)^k*(sum(C(k,j)*2^(1-j)*sum((-1)^((n-m)/2-i)*C(j,i)*(j-2*i)^(n-m+j)/(n-m+j)!, i=0..floor(j/2))*(-1)^(k-j), j=1..k))*C(k+n-1,n-1), k=1..n-m))/(m-1)!, m=1..n), n>0. - Vladimir Kruchinin, Sep 12 2010

E.g.f.: exp(arcsin(x))=1+2z/(H(0)-z); H(k)=4k+2+z^2*(4k^2+8k+5)/H(k+1), where z=x/((1-x^2)^1/2); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2011

a(n) ~ (exp(Pi/2)-(-1)^n*exp(-Pi/2)) * n^(n-1) / exp(n). - Vaclav Kotesovec, Oct 23 2013

a(n) = 2^(n-2) * (exp(Pi/2)-(-1)^n*exp(-Pi/2)) * GAMMA((n-I)/2) * GAMMA((n+I)/2) / Pi. - Vaclav Kotesovec, Nov 06 2014

Extensions

More terms from Christian G. Bower