A048602 -id:A048602 - cp's OEIS frontend

A048603 Denominators of coefficients in function a(x) such that a(a(x)) = sin x.

1, 12, 160, 40320, 71680, 1277337600, 79705866240, 167382319104000, 91055981592576000, 62282291409321984000, 4024394214140805120000, 5882770031248492462080000, 9076273762497674084352000000

Offset: 0

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

Also denominators of coefficients in function a(x) such that a(a(x)) = sinh x.

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

			x - x^3/12 - x^5/160 ...

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. A048602, A048606.

n = 13; 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] // Denominator // Partition[#, 2] &)[[All, 2]]
(* Jean-François Alcover, May 05 2011 *)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007

A052134 Numerators of coefficients in function a(x) such that a(a(a(x))) = sinh x.

Original entry on oeis.org

1, 1, -7, 643, -13583, 29957, -24277937, 6382646731, 2027394133729, -10948179003324221, 177623182156029053, -126604967848904128751, -2640658729595838040517543, 423778395125199663867841, 134802774190189008299452419971

Offset: 0

N. J. A. Sloane, Jan 22 2000

W. C. Yang, Composition equations, preprint, 1999.

Cf. A052135. See also A048602, A048603, etc.

Apart from signs, same as A052132?

n = 15; m (* = maximal degree *)= 2n - 1; a[x_] = Sum[c[k]*x^k, {k, 1, m, 2}]; coes = DeleteCases[ CoefficientList[ Series[ a@a@a@x - Sinh[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, Jan 04 2013 *)

More terms from Vladeta Jovovic, Jul 16 2002

A052135 Denominators of coefficients in function a(x) such that a(a(a(x))) = sin x.

Original entry on oeis.org

1, 18, 1080, 408240, 29393280, 215550720, 648499737600, 953294614272000, 2333665215737856000, 7183021534041120768000, 232066849561328517120000, 723095875143672151080960000, 74189636789740762700906496000000

Offset: 0

N. J. A. Sloane, Jan 22 2000

Also denominators of coefficients in function a(x) such that a(a(a(x))) = sinh x.

W. C. Yang, Composition equations, preprint, 1999.

Cf. A052132, A052134. See also A048602, A048603, etc.

n = 13; m = 2 n - 1 (* m = maximal degree *); a[x_] = Sum[c[k] x^k, {k, 1, m, 2}] ; coes = DeleteCases[ CoefficientList[Series[a @ 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] // Denominator // Partition[#, 2] &)[[All, 2]]
(* Jean-François Alcover, May 04 2011 *)

More terms from Vladeta Jovovic, Jul 16 2002

A052136 Numerators of power series coefficients of a(x) satisfying a(a(a(x)))= arctan(x).

Original entry on oeis.org

1, -1, 4, -263, 181, -19417, 2650183, -334415182, 2505796264, -1075533383968, 644250947168711, -35934792935656882, 59703596150692742866, -2784264154855168826899, 13245106337447512956269, 145404446885533849363819862, -576405412549008975387674250194

Offset: 0

N. J. A. Sloane, Jan 22 2000

W. C. Yang, Composition equations, preprint, 1999.

Cf. A052137. See also A048602, A048603, etc.

interface(labeling=false) : a := 0 : mPow := 17 : for i from 0 to mPow do a := a+alph[2*i+1]*x^(2*i+1) ; od: a2 := 0 : for i from 0 to mPow do a2 := a2+alph[2*i+1]*a^(2*i+1) ; od: a2 := taylor(a2,x=0,2*mPow+2) : a2 := convert(a2,polynom) : a3 := 0 : for i from 0 to mPow do a3 := a3+alph[2*i+1]*a2^(2*i+1) ; od: for i from 0 to mPow do tanCoef[2*i+1] := coeftayl(arctan(x),x=0,2*i+1) ; od: a3 := taylor(a3,x=0,2*mPow+2) : a3 := convert(a3,polynom) : for i from 0 to mPow do tozer := coeftayl(a3,x=0,2*i+1) : alph[2*i+1] := op(1,[solve(tozer=tanCoef[2*i+1],alph[2*i+1])]) : printf("%d, ",numer(alph[2*i+1])) ; ; od: # R. J. Mathar, Jun 21 2007

n = 17; m = 2n - 1 (* m = maximal degree *);
a[x_] = Sum[c[k] x^k, {k, 1, m, 2}] ; coes = DeleteCases[ CoefficientList[ Series[a @ a @ a @ x - ArcTan[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 16 2011 *)
T[n_, n_] = 1; T[n_, m_] := T[n, m] = 1/3*(2^(-m - 1)*m!*((-1)^(n + m) + 1)*(-1)^((3*n + m)/2)*Sum[2^i*StirlingS1[i, m]*Binomial[n-1, i-1]/i!, {i, m, n}] - Sum[T[k, m]*Sum[T[n, i]*T[i, k], {i, k, n}], {k, m+1, n-1}] - T[m, m]*Sum[T[n, i]*T[i, m], {i, m+1, n-1}]);
Table[T[2*n - 1, 1] // Numerator, {n, 1, 17}] (* Jean-François Alcover, Jul 13 2016, after Vladimir Kruchinin *)

T(n,m):=if n=m then 1 else 1/3*(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(k,m)*sum(T(n,i)*T(i,k),i,k,n),k,m+1,n-1)-T(m,m)*sum(T(n,i)*T(i,m),i,m+1,n-1));
makelist(num(T(2*n-1,1)),n,1,7); /* Vladimir Kruchinin, Mar 10 2012 */

a(x)=sum_{n=0,1,2,3...} A052136(n)/A052137(n)*x^(2n+1). - R. J. Mathar, Jun 21 2007

a(n)=numerator(T(2*n-1,1)), T(n,m)=1/3*(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(k=m+1..n-1, T(k,m)*sum(i=k..n, T(n,i)*T(i,k))) -T(m,m)*sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - Vladimir Kruchinin, Mar 10 2012]

More terms from R. J. Mathar, Jun 21 2007

A048606 Numerators of coefficients in function a(x) such that a(a(x)) = sinh(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)

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

			x + x^3/12 - x^5/160 + ...

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

W. C. Yang, Derivatives are essentially integer partitions, Discrete Math., 222 (2000), 235-245.

Cf. A048603. Apart from signs, the same sequence as A048602.

A052132 Numerators of coefficients in function a(x) such that a(a(a(x))) = sin x.

Original entry on oeis.org

1, -1, -7, -643, -13583, -29957, -24277937, -6382646731, 2027394133729, 10948179003324221, 177623182156029053, 126604967848904128751, -2640658729595838040517543, -423778395125199663867841

Offset: 0

N. J. A. Sloane, Jan 22 2000

W. C. Yang, Composition equations, preprint, 1999.

Cf. A052135. See also A048602, A048603, etc.

Apart from signs, same as A052134?

n = 14; m = 2 n - 1 (* m = maximal degree *); a[x_] = Sum[c[k] x^k, {k, 1, m, 2}] ; coes = DeleteCases[ CoefficientList[Series[a @ 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 04 2011 *)

T(n,m):=if n=m then 1 else 1/3*((((-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(k,m)*sum(T(n,i)*T(i,k),i,k,n),k,m+1,n-1)-T(m,m)*sum(T(n,i)*T(i,m),i,m+1,n-1));
makelist(num(T(2*n-1,1)),n,1,7); /* Vladimir Kruchinin, Mar 10 2012 */

a(n) = numerator(T(2*n-1,1)), T(n,m) = 1/3*((((-1)^(n-m)+1)*sum(i=0..m/2, (2*i-m)^n*binomial(m,i)*(-1)^((n+m)/2-i)))/(2^m*n!)-sum(k=m+1..n-1, T(k,m)*sum(i=k..n, T(n,i)*T(i,k)))-T(m,m)*sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - Vladimir Kruchinin, Mar 10 2012

More terms from R. J. Mathar, coded equivalent to A052136 - R. J. Mathar, Dec 09 2009

A052137 Denominators of power series coefficients of a(x) satisfying a(a(a(x)))= arctan(x).

Original entry on oeis.org

1, 9, 135, 25515, 45927, 12629925, 4433103675, 1396427657625, 23739270179625, 21920842083865725, 34525326282088516875, 8734907549368394769375, 17688187787470999407984375, 413903594226821386146834375

Offset: 0

N. J. A. Sloane, Jan 22 2000

W. C. Yang, Composition equations, preprint, 1999.

Cf. A052136. See also A048602, A048603, etc.

interface(labeling=false) : a := 0 : mPow := 15 : for i from 0 to mPow do a := a+alph[2*i+1]*x^(2*i+1) ; od: a2 := 0 : for i from 0 to mPow do a2 := a2+alph[2*i+1]*a^(2*i+1) ; od: a2 := taylor(a2,x=0,2*mPow+2) : a2 := convert(a2,polynom) : a3 := 0 : for i from 0 to mPow do a3 := a3+alph[2*i+1]*a2^(2*i+1) ; od: for i from 0 to mPow do tanCoef[2*i+1] := coeftayl(arctan(x),x=0,2*i+1) ; od: a3 := taylor(a3,x=0,2*mPow+2) : a3 := convert(a3,polynom) : for i from 0 to mPow do tozer := coeftayl(a3,x=0,2*i+1) : alph[2*i+1] := op(1,[solve(tozer=tanCoef[2*i+1],alph[2*i+1])]) : printf("%d, ",denom(alph[2*i+1])) ; ; od: # R. J. Mathar, Jun 21 2007

n = 14; m = 2 n - 1 (* m = maximal degree *);
a[x_] = Sum[c[k] x^k, {k, 1, m, 2}]; coes = DeleteCases[ CoefficientList[ Series[a @ a @ a @ x - ArcTan[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] // Denominator // Partition[#, 2] &)[[All, 2]] (* Jean-François Alcover, May 16 2011 *)
T[n_, n_] = 1; T[n_, m_] := T[n, m] = 1/3*(2^(-m - 1)*m!*((-1)^(n + m) + 1)*(-1)^((3*n + m)/2)*Sum[2^i*StirlingS1[i, m]*Binomial[n - 1, i - 1]/i!, {i, m, n}] - Sum[T[k, m]*Sum[T[n, i]*T[i, k], {i, k, n}], {k, m + 1, n - 1}] - T[m, m]*Sum[T[n, i]*T[i, m], {i, m + 1, n - 1}]);
Table[T[2*n - 1, 1] // Denominator, {n, 1, 14}] (* Jean-François Alcover, Jul 13 2016, after Vladimir Kruchinin *)

a(x) = Sum_{n>=0} (A052136(n)/A052137(n))*x^(2n+1). - R. J. Mathar, Jun 21 2007

More terms from R. J. Mathar, Jun 21 2007

A052138 Numerators of coefficients in function a(x) such that a(a(a(x))) = log (1+x).

Original entry on oeis.org

1, -1, 1, -5, 103, -49, 2971, -34409, 10787, -567923, 4857119, -30312479, 7045653829, -77510407993, 262952596463, -58196505296117, 74362625639717, 172749391066639, -24672728226124829, -12265872397466432881, 1772477431815925044131

Offset: 1

N. J. A. Sloane, Jan 22 2000

A(x)=sum(n>0 b(n)x^n/(3^(n-1)*n!), b(n)=T(n,1)*n!*3^(n-1) - integer.

W. C. Yang, Composition equations, preprint, 1999.

Cf. A052139. See also A048602, A048603, etc.

max = 21; f[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 0; c[1] = 1; s[1] = {}; coes = CoefficientList[ Series[ f[f[f[x]]] - Log[1 + x], {x, 0, max}], x]; eqns = Rest[ Thread[coes == 0]]; Do[eqns = Rest[eqns] /. s[k]; s[k+1] = Solve[eqns[[1]], c[k + 1]][[1]], {k, 1, max-1}]; Numerator[ Table[c[k], {k, 1, max}] /. Flatten[ Table[s[k], {k, 1, max}]]] (* Jean-François Alcover, Oct 19 2011 *)
T[n_, m_] := T[n, m] = If[n==m, 1, 1/3*(StirlingS1[n, m]*m!/n! - Sum[T[k, m]*Sum[T[n, i]*T[i, k], {i, k, n}], {k, m+1, n-1}] - T[m, m]*Sum[T[n, i]*T[i, m], {i, m+1, n-1}])]; Table[T[n, 1] // Numerator, {n, 1, 21}] (* Jean-François Alcover, Dec 15 2015, after Vladimir Kruchinin *)

T(n,m):=if n=m then 1 else 1/3*(stirling1(n,m)*m!/n!-sum(T(k,m)*sum(T(n,i)*T(i,k),i,k,n),k,m+1,n-1)-T(m,m)*sum(T(n,i)*T(i,m),i,m+1,n-1));
makelist(num(T(n,1)),n,1,7); /* Vladimir Kruchinin, Mar 10 2012 */

a(x)=sum_{n=1,2,3..} A052138(n)/A052139(n)*x^n. - R. J. Mathar, Jun 21 2007

a(n):=numerator(T(n,1)), T(n,m)=1/3*(stirling1(n,m)*m!/n!-sum(k=m+1..n-1, T(k,m)*sum(i=k..n, T(n,i)*T(i,k)))-T(m,m)*sum(i,m+1,n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - Vladimir Kruchinin, Mar 10 2012

More terms from R. J. Mathar, Jun 21 2007

A052139 Denominators of coefficients in function a(x) such that a(a(a(x))) = log (1+x).

Original entry on oeis.org

1, 6, 18, 216, 9720, 9720, 1224720, 29393280, 18895680, 1984046400, 32736765600, 392841187200, 183849675609600, 4211828932147200, 25738954585344000, 8339421285651456000, 21265524278411212800, 437462213727316377600

Offset: 1

N. J. A. Sloane, Jan 22 2000

W. C. Yang, Composition equations, preprint, 1999.

Cf. A052138. See also A048602, A048603, etc.

max = 18; f[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 0; c[1] = 1; s[1] = {}; coes = CoefficientList[ Series[ f[f[f[x]]] - Log[1 + x], {x, 0, max}], x]; eqns = Rest[ Thread[ coes == 0]]; Do[eqns = Rest[eqns] /. s[k]; s[k+1] = Solve[eqns[[1]], c[k + 1]][[1]], {k, 1, max-1}]; Denominator[ Table[c[k], {k, 1, max}] /. Flatten[ Table[s[k], {k, 1, max}]]] (* Jean-François Alcover, Oct 19 2011 *)
T[n_, m_] := T[n, m] = If[n == m, 1, 1/3*(StirlingS1[n, m]*m!/n! - Sum[T[k, m]*Sum[T[n, i]*T[i, k], {i, k, n}], {k, m+1, n-1}] - T[m, m]*Sum[T[n, i]* T[i, m], {i, m+1, n-1}])]; Table[T[n, 1] // Denominator, {n, 1, 21}] (* Jean-François Alcover, Dec 15 2015, after Vladimir Kruchinin *)

a(x)=sum_{n=1,2,3..} A052138(n)/A052139(n)*x^n. - R. J. Mathar, Jun 21 2007

More terms from R. J. Mathar, Jun 21 2007

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

A048603 Denominators of coefficients in function a(x) such that a(a(x)) = sin x.

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A052134 Numerators of coefficients in function a(x) such that a(a(a(x))) = sinh x.

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A052135 Denominators of coefficients in function a(x) such that a(a(a(x))) = sin x.

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A052136 Numerators of power series coefficients of a(x) satisfying a(a(a(x)))= arctan(x).

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A048606 Numerators of coefficients in function a(x) such that a(a(x)) = sinh(x).

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A052132 Numerators of coefficients in function a(x) such that a(a(a(x))) = sin x.

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A052137 Denominators of power series coefficients of a(x) satisfying a(a(a(x)))= arctan(x).

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Maple

Mathematica

Formula

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A052138 Numerators of coefficients in function a(x) such that a(a(a(x))) = log (1+x).

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