A006228 Expansion of exp(arcsin(x)). (Formerly M1523)

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

0,4

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.

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

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 *)

Prog

(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 */

Crossrefs

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

Bisections are A101927 and A101928.

Cf. A002019.

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

Sequence in context: A192101 A012768 A170947 * A363140 A190656 A262166

Adjacent sequences: A006225 A006226 A006227 * A006229 A006230 A006231

Keyword

nonn

Author

N. J. A. Sloane, Jeffrey Shallit

Extensions

More terms from Christian G. Bower

Status

approved