A095885 -id:A095885 - cp's OEIS frontend

A095883 Let F(x) be the function such that F(F(x)) = arcsin(x), then F(x) = Sum_{n>=0} a(n)/2^n*x^(2n+1)/(2n+1)!.

1, 1, 13, 501, 38617, 4945385, 944469221, 250727790173, 88106527550129, 39555449833828817, 22093952731139969213, 15041143328788464370373, 12273562321018687866908553, 11833097802606125967312406457

Offset: 0

Paul D. Hanna, Jun 10 2004

It appears that, if arcsin(x) is changed to arcsinh(x) in the definition, the sequence obtained is the same except alternating in sign: 1, -1, 13, -501, ... - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Jul 16 2009

a(35) is negative. - Vaclav Kotesovec, Jan 06 2023

			F(x) = x + (1/2)*x^3/3! + (13/2^2)*x^5/5! + (501/2^3)*x^7/7! + (38617/2^4)*x^9/9! + ...
Special values:
F(x)=Pi/6 at x=F(1/2) = 0.51137532057552418592144885355...
F(x)=Pi/4 at x=F(sqrt(2)/2) = 0.74287348600976...

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A095882, A095884, A095885.

a[n_] := Module[{A, B, F}, F = ArcSin[x] + O[x]^(2n+3); A = F; For[i = 0, i <= n, i++, B[x_] = InverseSeries[A, x] // Normal; A = (A + B[F])/2]; 2^n* (2n+1)!*SeriesCoefficient[A, {x, 0, 2n+1}]];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Aug 16 2022, after PARI code *)

{a(n)=local(A,B,F);F=asin(x+x*O(x^(2*n+1)));A=F; for(i=0,n,B=serreverse(A);A=(A+subst(B,x,F))/2);2^n*(2*n+1)!*polcoeff(A,2 *n+1,x)}

a(n)=T(2*n+1,1)*2^n*(2*n+1)!, T(n,m)=if n=m then 1 else 1/2(Co(n,m)-sum(i=m+1..n-1, T(n,i)*T(i,m))), Co(n,m)=T121408(n,m)=(m!*(sum(k=0..n-m, (-1)^((k)/2)*(sum(i=0..k, (2^i*stirling1(m+i,m)* binomial(m+k-1,m+i-1))/(m+i)!))*binomial((n-2)/2,(n-m-k)/2)))*((-1)^(n-m)+1))/2. - Vladimir Kruchinin, Nov 11 2011

A048605 Numerators of coefficients in function a(x) such that a(a(x)) = arctan(x).

Original entry on oeis.org

1, -1, 7, -43, 4489, -49897, 20130311, -319053131, 329796121169, -62717244921977, 14635852695795623, -33233512260583073, 149490010959849868177, -3562767949848393597053

Offset: 0

Winston C. Yang (yang(AT)math.wisc.edu)

A recursion exists for coefficients, but is too complicated to use without a computer algebra system.

			x - x^3/6 + x^5 * 7/120 + ...

W. C. Yang, Polynomials are essentially integer partitions, preprint, 1999
W. C. Yang, Composition equations, preprint, 1999

Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x) = F(x), arXiv:1302.1986
W. C. Yang, Derivatives are essentially integer partitions, Discrete Math., 222 (2000), 235-245.

Cf. A048604, A095885.

n = 28; a[x_] = Sum[c[k] k! x^k, {k, 1, n, 2}];
sa = Series[a[x], {x, 0, n}];
coes = CoefficientList[ComposeSeries[sa, sa] - Series[ArcTan[x], {x, 0, n}], x] // Rest;
eq = Reduce[((# == 0) & /@ coes)]; Table[c[k] k!, {k, 1, n, 2}] /. First[Solve[eq]] // Numerator
(* Jean-François Alcover, Apr 26 2011 *)

T(n, m):=if n=m then 1 else 1/2*(2^(-m-1)*m!*((-1)^(n+m)+1)*(-1)^((3*n+m)/2)*sum((2^i*stirling1(i, m)*binomial(n-1, i-1))/i!, i, m, n)-sum(T(n, i)*T(i, m), i, m+1, n-1));
makelist(num(T(2*n-1, 1), n, 1, 5)); /* Vladimir Kruchinin, Mar 12 2012 */

a(n) = numerator(T(2*n-1,1)), T(n,m)=1/2*(2^(-m-1)*m!*((-1)^(n+m)+1)*(-1)^((3*n+m)/2)*sum(i=m..n, (2^i*stirling1(i,m)*binomial(n-1,i-1))/i!)-sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - Vladimir Kruchinin, Mar 12 2012

A072350 E.g.f. A(x) satisfies A(A(x)) = tan(x), where A(x) = Sum_{n>=1} a(n)*x^(2n-1)/(2n-1)!.

Original entry on oeis.org

1, 1, 3, 17, 225, 3613, -42997, 8725357, 2116966081, -549193907111, -114757574954509, 117893333517545097, 14433599120070484321, -65568697910890921624715, 2968238619232726100394235, 86999609037195113208781248165

Offset: 1

Vladeta Jovovic, Jul 17 2002

sign

The inverse of this g.f. A(x) is the g.f. of A095885. - Paul D. Hanna, Dec 09 2004

			a(x) = x/1!+x^3/3!+3*x^5/5!+17*x^7/7!+225*x^9/9!+3613*x^11/11!-42997*x^13/13!+...

Paul D. Hanna, Table of n, a(n) for n = 1..40
Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013.

Cf. A048602, A048603, A052134, A052135.

Cf. A095885 (inverse).

a[n_] := Module[{A, B, F}, F = Tan[x + O[x]^(2n+1)]; A = F; For[i = 0, i <= 2n-1, i++, B = InverseSeries[A, x]; A = (A + (B /. x -> F))/2]; If[n<1, 0, (2n-1)!*SeriesCoefficient[A, {x, 0, 2n-1}]]]; Table[a[n], {n, 1, 16}] (* Jean-François Alcover, Oct 29 2015, adapted from PARI *)

{a(n)=local(A,B,F);F=tan(x+O(x^(2*n+1)));A=F; for(i=0,2*n-1,B=serreverse(A);A=(A+subst(B,x,F))/2); if(n<1,0,(2*n-1)!*polcoeff(A,2*n-1,x))} \\ Paul D. Hanna, Dec 09 2004

a(n)=(2*n-1)!*T(2*n-1,1), T(n,k)=if n=k then 1 else 1/2*(T059419(n,k)*k!/n!-sum(i=k+1..n-1, T(n,i)*T(i,k))). [Vladimir Kruchinin, Nov 11 2011]

More terms from Paul D. Hanna, Dec 09 2004

cp's OEIS Frontend

A095883 Let F(x) be the function such that F(F(x)) = arcsin(x), then F(x) = Sum_{n>=0} a(n)/2^n*x^(2n+1)/(2n+1)!.

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A048605 Numerators of coefficients in function a(x) such that a(a(x)) = arctan(x).

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A072350 E.g.f. A(x) satisfies A(A(x)) = tan(x), where A(x) = Sum_{n>=1} a(n)*x^(2n-1)/(2n-1)!.

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