A048602
Numerators of coefficients in function a(x) such that a(a(x)) = sin(x).
Original entry on oeis.org
1, -1, -1, -53, -23, -92713, -742031, 594673187, 329366540401, 104491760828591, 1508486324285153, -582710832978168221, -1084662989735717135537, -431265609837882130202597, 784759327625761394688977441
Offset: 0
Winston C. Yang (yang(AT)math.wisc.edu)
- W. C. Yang, Polynomials are essentially integer partitions, preprint, 1999
- W. C. Yang, Composition equations, preprint, 1999
Apart from signs, the same sequence as
A048606.
-
n = 15; m = 2 n - 1 (* m = maximal degree *); a[x_] = Sum[c[k] x^k, {k, 1, m, 2}] ; coes = DeleteCases[CoefficientList[Series[a@a@x - Sin[x], {x, 0, m}], x] // Rest , 0]; Do[s[k] = Solve[coes[[1]] == 0] // First; coes = coes /. s[k] // Rest, {k, 1, n}]; (- CoefficientList[a[x] /. Flatten @ Array[s, n], x] // Numerator // Partition[#, 2] &)[[All, 2]] (* Jean-François Alcover, May 05 2011 *)
-
T(n,m):= if n=m then 1 else ((((-1)^(n-m)+1)*sum((2*i-m)^n*binomial(m,i)*(-1)^((n+m)/2-i),i,0,m/2))/(2^m*n!)-sum(T(n,i)*T(i,m),i,m+1,n-1))/2; makelist(num(T(n,1)),n,1,10); /* Vladimir Kruchinin, Nov 08 2011 */
-
a(n) = { my(ps = sin(x + O(x^(2*n))), q=0); while(ps<>q, q=ps; ps=(sin(serreverse(ps)) + ps)/2); numerator(polcoef(ps, 2*n-1)) } \\ Gottfried Helms, Feb 20 2022
A280795
E.g.f. A(x) = F(x)^2, where F(F(x)) = sin(x).
Original entry on oeis.org
2, -4, -4, -64, -1392, -36048, -68288, 217764736, 45152039424, 6761531610496, 479656304761728, -181164506307765760, -94510602811595547648, -9099359091145965684736, 15760449987522859047213056, 8420566548360384773538111488, -5361092182457081839427169370112, -9833526816807364784720991628738560, 595009928118250648162581567581609984
Offset: 1
E.g.f.: A(x) = 2*x^2/2! - 4*x^4/4! - 4*x^6/6! - 64*x^8/8! - 1392*x^10/10! - 36048*x^12/12! - 68288*x^14/14! + 217764736*x^16/16! + 45152039424*x^18/18! + 6761531610496*x^20/20! +...
where A(x) = F(x)^2 such that F(F(x)) = sin(x) and F(x) begins:
F(x) = x - 1/2*x^3/3! - 3/4*x^5/5! - 53/8*x^7/7! - 1863/16*x^9/9! - 92713/32*x^11/11! - 3710155/64*x^13/13! + 594673187/128*x^15/15! + 329366540401/256*x^17/17! + 104491760828591/512*x^19/19! + 19610322215706989/1024*x^21/21! +...
-
{a(n)=local(A, B, F); F=sin(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)!*polcoeff(A^2, 2*n, x))}
for(n=1, 30, print1(a(n), ", "))
A280796
E.g.f. sqrt(1 - F(x)^2), where F(F(x)) = sin(x).
Original entry on oeis.org
1, -1, -1, -13, -367, -18549, -1465953, -166262441, -25526158559, -5088504559209, -1277203721141441, -394351281523218693, -147069662568684159055, -65255038602423680990301, -33992871584988519888865825, -20539580068386370855911967393, -14240009948449682448965044873663, -11224227759618581623496389602591953, -9984932631658989030110444663072663937, -9963529315662216629464111409263738683133
Offset: 0
E.g.f.: A(x) = 1 - x^2/2! - x^4/4! - 13*x^6/6! - 367*x^8/8! - 18549*x^10/10! - 1465953*x^12/12! - 166262441*x^14/14! - 25526158559*x^16/16! - 5088504559209*x^18/18! - 1277203721141441*x^20/20! - 394351281523218693*x^22/22! - 147069662568684159055*x^24/24! - 65255038602423680990301*x^26/26! +...
such that A(x) = sqrt(1 - F(x)^2) where F(F(x)) = sin(x) and F(x) begins:
F(x) = x - 1/2*x^3/3! - 3/4*x^5/5! - 53/8*x^7/7! - 1863/16*x^9/9! - 92713/32*x^11/11! - 3710155/64*x^13/13! + 594673187/128*x^15/15! + 329366540401/256*x^17/17! + 104491760828591/512*x^19/19! + 19610322215706989/1024*x^21/21! - 5244397496803513989/2048*x^23/23! - 7592640928150019948759/4096*x^25/25! +...
Also, e.g.f. A(x) satisfies: A(F(x)) = cos(x).
-
{a(n)=local(A, B, F); F=sin(x+O(x^(2*n+1))); A=F; for(i=0, 2*n-1, B=serreverse(A); A=(A+subst(B, x, F))/2); (2*n)!*polcoeff(sqrt(1-A^2), 2*n, x)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.
Comments