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A006228
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Expansion of exp(arcsin x).
(Formerly M1523)
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10
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1, 1, 1, 2, 5, 20, 85, 520, 3145, 26000, 204425, 2132000, 20646925, 260104000, 2993804125, 44217680000, 589779412625, 9993195680000, 151573309044625, 2898026747200000, 49261325439503125, 1049085682486400000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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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).
H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565
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FORMULA
| i even: a_i = prod_{j=1}^{i/2-1} 1 + 4j^2, i odd: a_i = prod_{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 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 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(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), n>0. [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), 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
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MATHEMATICA
| Distribute[ CoefficientList[ Series[ E^ArcSin[x], {x, 0, 21}], x] * Table[ n!, {n, 0, 21}]] (from Robert G. Wilson v Feb 10 2004)
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PROG
| (Other) 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); (for Maxima) [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Sep 12 2010]
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CROSSREFS
| Bisections are expansions of sin(asinh(x)) and cos(asinh(x)).
Bisections are A101927 and A101928.
Cf. A002019.
Cf. A166741, A166748. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 24 2009]
Sequence in context: A192101 A012768 A170947 * A190656 A002484 A003069
Adjacent sequences: A006225 A006226 A006227 * A006229 A006230 A006231
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jeffrey Shallit
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EXTENSIONS
| More terms from Christian G. Bower (bowerc(AT)usa.net)
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