cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A143598 E.g.f.: A(x) = exp(x*sinh(x*G(x))) where G(x) = cosh(x*G(x)) is the e.g.f. of A143601.

Original entry on oeis.org

1, 2, 28, 1176, 103440, 15726880, 3684098496, 1232799974784, 558670427013376, 329559835063067136, 245462725323910487040, 225319148634038399801344, 249936012383478860884217856, 329609037187846742271984869376
Offset: 0

Views

Author

Paul D. Hanna, Aug 27 2008

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x^2/2! + 28*x^4/4! + 1176*x^6/6! + 103440*x^8/8! +...
A(x) = exp(x*F(x)) where F(x) = e.g.f. of A007106:
F(x) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! +...
A(x) = exp(x*sqrt(G(x)^2 - 1)) where G(x) = e.g.f. of A143601:
G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
A(x) = sqrt(H(x)*H(-x)) where H(x) = e.g.f. of A143599:
H(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 316*x^5/5! +...

Crossrefs

Cf. A058014, A143600, A143601, A007106.

Programs

  • PARI
    {a(n)=local(G=1+x*O(x^n));for(i=0,n,G=cosh(x*G));n!*polcoeff(exp(x*sqrt(G^2-1)),n)}

Formula

E.g.f.: A(x) = exp(x*F(x)) where F(x) is the e.g.f. of A007106.
E.g.f.: A(x) = sqrt(H(x)*H(-x)) where H(x) = exp(x*sqrt(H(x)/H(-x))) is the e.g.f. of A143599.
E.g.f. satisfies: A(x/cosh(x)) = exp(x*tanh(x)). [From Paul D. Hanna, Aug 29 2008]

A007106 Number of labeled odd degree trees with 2n nodes.

Original entry on oeis.org

1, 4, 96, 5888, 686080, 130179072, 36590059520, 14290429935616, 7405376630685696, 4917457306800619520, 4071967909087792857088, 4113850542422629363482624, 4980673081258443273955966976, 7119048451600750435732824260608, 11861520124846917915630931846103040
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _Peter Bala_, Apr 24 2012: (Start)
Let G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + ... be the e.g.f. for A143601. Then sinh(x*G(x)) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + ....
Conjectural e.g.f. as an x-adic limit:
sinh(x) = x + ...; sinh(x*cosh(x)) = x + 4*x^3/3! + ...;
sinh(x*cosh(x*cosh(x))) = x + 4*x^3/3! + 96*x^5/5! + ...;
sinh(x*cosh(x*cosh(x*cosh(x)))) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + ....
(End)

References

  • R. W. Robinson, personal communication.
  • R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A036778, A058014, A060279, A143601, A156289.

Programs

  • Maple
    A007106(n) = A(2n) where n>=2, A(n) = (add(binomial(n,q)*(n-2*q)^(n-2)/(n-2)!, q=0..n) - add(binomial(n-1,q)*(n-2*q)^(n-3)/(n-3)!, q=0..n-1) + add(binomial(n-1,q)*(n-2-2*q)^(n-3)/(n-3)!, q=0..n-1))*n!/2^(n+1)/(n-1)
  • Mathematica
    {1}~Join~Array[(1/2)*Sum[Binomial[2 #, k]*(# - k)^(2 # - 2), {k, 0, # - 1}] &, 12, 2] (* Michael De Vlieger, Oct 13 2021 *)
  • PARI
    a(n) = if(n<=1, n==1, sum(k=0, n-1, binomial(2*n,k) * (n-k)^(2*n-2))/2) \\ Andrew Howroyd, Nov 22 2021

Formula

a(n) = A060279(n)/(2*n). - Vladeta Jovovic, Feb 08 2005
Bisection of A058014. The expansion 1/sqrt(1+x^2)*arcsinh(x) = x - 4*x^3/3! + 64*x^5/5! - ... (see A002454) has series reversion x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + .... The coefficients appear to be the terms of this sequence. As an x-adic limit this e.g.f. equals lim_{n -> infinity} sinh(f(n,x)), where f(0,x) = x and f(n,x) = x*cosh(f(n-1,x)) for n >= 1. See the example section below. - Peter Bala, Apr 24 2012
a(n) = Sum_{k=1..n} binomial(n,k) * k! * (n-2)! [z^{n-2}] [u^k] exp(u(exp(z)+exp(-z)-2)/2)). - Marko Riedel, Jun 16 2016
From Alexander Burstein, Oct 13 2021: (Start)
a(n) = (1/2) * Sum_{k=0..n-1} binomial(2*n,k) * (n-k)^(2*n-2) for n >= 2.
a(n) = (2*n-1)!*[x^(2*n-1)] sinh(REVERT(x/cosh(x))), see A036778. (End)
a(n) = Sum_{k=0..n-1} A156289(n-1, k)*(2*n)!/(2*n-k)!. - Peter Luschny, May 07 2022

Extensions

Corrected and extended by Vladeta Jovovic, Feb 08 2005

A058014 Number of labeled trees with n+1 nodes such that the degrees of all nodes, excluding the first node, are odd.

Original entry on oeis.org

1, 1, 1, 4, 13, 96, 541, 5888, 47545, 686080, 7231801, 130179072, 1695106117, 36590059520, 567547087381, 14290429935616, 257320926233329, 7405376630685696, 151856004814953841, 4917457306800619520, 113144789723082206461, 4071967909087792857088
Offset: 0

Views

Author

Alex Postnikov (apost(AT)math.mit.edu), Nov 13 2000

Keywords

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 4x^3/3! + 13x^4/4! + 96x^5/5! +...

Links

Crossrefs

Cf. bisections: A007106, A143601.
Cf. A138764 (variant).

Programs

  • Maple
    a := n -> 2^(-n)*add(binomial(n,k)*(n+1-2*k)^(n-1), k=0..n);
  • Mathematica
    a[n_] := Sum[((n-2k+1)^(n-1)*n!) / (k!*(n-k)!), {k, 0, n}] / 2^n; a[1] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 14 2011, after Maple *)
  • PARI
    {a(n)=local(A=1+x);for(i=0,n,A=exp(x*(A+1/(A +x*O(x^n)))/2));n!*polcoeff(A,n)} \\ Paul D. Hanna, Mar 29 2008
  • PARI
    {a(n) = sum(k=0, n, binomial(n, k)*(n+1-2*k)^(n-1))/2^n} \\ Seiichi Manyama, Sep 27 2020

Formula

a(n) = (1/2^n) * Sum_{k=0..n} binomial(n,k) * (n + 1 - 2*k)^(n-1).
From Paul D. Hanna, Mar 29 2008: (Start)
E.g.f. satisfies A(x) = exp( x*[A(x) + 1/A(x)]/2 ).
E.g.f. A(x) equals the inverse function of 2*x*log(x)/(1 + x^2).
Let r = radius of convergence of A(x), then r = 0.6627434193491815809747420971092529070562335491150224... and A(r) = 3.31905014223729720342271370055697247448941708369151595... where A(r) and r satisfy A(r) = exp( (A(r)^2 + 1)/(A(r)^2 - 1) ) and r = 2*A(r)/(A(r)^2 - 1). (End)
E.g.f. A(x)=exp(B(x)), B(x) satisfies B(x)=x*cosh(B(x)). [Vladimir Kruchinin, Apr 19 2011]
a(n) ~ (1-(-1)^n*s^2)/s * n^(n-1) * ((1-s^2)/(2*s))^n / exp(n), where s = 0.3012910191606573456... is the root of the equation (1+s^2) = (s^2-1)*log(s), r = 2*s/(1-s^2). - Vaclav Kotesovec, Jan 08 2014
E.g.f. satisfies A(-x) = 1/A(x). - Alexander Burstein, Oct 26 2021
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (2*k+1) * binomial(n-1,2*k) * a(2*k) * a(n-1-2*k). - Seiichi Manyama, Jul 05 2025

Extensions

Updated URL and author's e-mail address - R. J. Mathar, May 23 2010

A370430 Expansion of e.g.f. C(x,k) satisfying C(x,k) = cosh( x*cosh(k*x*C(x,k)) ), as a triangle read by rows.

Original entry on oeis.org

1, 1, 0, 1, 12, 0, 1, 420, 120, 0, 1, 10248, 36400, 896, 0, 1, 196920, 4858560, 2170560, 5760, 0, 1, 3247860, 461126160, 1127738304, 102960000, 33792, 0, 1, 48361404, 35248293080, 340884800256, 187282263168, 4083183104, 186368, 0, 1, 669616080, 2290777550880, 76526954183680, 153279541958400, 25081621813248, 141360128000, 983040, 0
Offset: 0

Views

Author

Paul D. Hanna, Feb 19 2024

Keywords

Comments

The row sums equal A143601, the number of labeled odd degree trees with 2n+1 nodes.
Unsigned version of triangle A370330.
A row reversal of triangle A370432.

Examples

			E.g.f.: C(x,k) = 1 + (1)*x^2/2! + (1 + 12*k^2)*x^4/4! + (1 + 420*k^2 + 120*k^4)*x^6/6! + (1 + 10248*k^2 + 36400*k^4 + 896*k^6)*x^8/8! + (1 + 196920*k^2 + 4858560*k^4 + 2170560*k^6 + 5760*k^8)*x^10/10! + (1 + 3247860*k^2 + 461126160*k^4 + 1127738304*k^6 + 102960000*k^8 + 33792*k^10)*x^12/12! + (1 + 48361404*k^2 + 35248293080*k^4 + 340884800256*k^6 + 187282263168*k^8 + 4083183104*k^10 + 186368*k^12)*x^14/14! + ...
where C(x,k) = cosh( x*cosh(k*x*C(x,k)) ).
This triangle of coefficients a(n,j) of x^(2*n)*k^(2*j)/(2*n)! in C(x,k) begins
 1;
 1, 0;
 1, 12, 0;
 1, 420, 120, 0;
 1, 10248, 36400, 896, 0;
 1, 196920, 4858560, 2170560, 5760, 0;
 1, 3247860, 461126160, 1127738304, 102960000, 33792, 0;
 1, 48361404, 35248293080, 340884800256, 187282263168, 4083183104, 186368, 0;
 1, 669616080, 2290777550880, 76526954183680, 153279541958400, 25081621813248, 141360128000, 983040, 0;
 1, 8781531696, 131249560881600, 14052066349007232, 83205186217021440, 51607880705931264, 2855197025501184, 4416170065920, 5013504, 0; ...

Links

Crossrefs

Cf. A370431 (S), A370432 (D), A370433 (T), A143601 (row sums).
Cf. A370330.

Programs

  • PARI
    {a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cosh( x*cosh(k*x*C +Ox) );
S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
D = cosh( k*x*cosh(x*D +Ox));
T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))););
(2*n)! * polcoeff(polcoeff(C, 2*n, x), 2*j, k)}
for(n=0, 10, for(j=0, n, print1( a(n, j), ", ")); print(""))

Formula

E.g.f.: C(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n)*k^(2*j)/(2*n)! along with the related functions C = C(x,k), S = S(x,k), D = D(x,k), and T = T(x,k) satisfy the following formulas.
Definition.
(1.a) (C + S) = exp(x*D).
(1.b) (D + k*T) = exp(k*x*C).
(2.a) C^2 - S^2 = 1.
(2.b) D^2 - k^2*T^2 = 1.
Hyperbolic functions.
(3.a) C = cosh(x*D).
(3.b) S = sinh(x*D).
(3.c) D = cosh(k*x*C).
(3.d) T = (1/k) * sinh(k*x*C).
(4.a) C = cosh( x*cosh(k*x*C) ).
(4.b) S = sinh( x*cosh(k*x*sqrt(1 + S^2)) ).
(4.c) D = cosh( k*x*cosh(x*D) ).
(4.d) T = (1/k) * sinh( k*x*cosh(x*sqrt(1 + k^2*T^2)) ).
(5.a) (C*D + k*S*T) = cosh(x*D + k*x*C).
(5.b) (S*D + k*C*T) = sinh(x*D + k*x*C).
Transformations.
(6.a) C(x, 1/k) = D(x/k, k).
(6.b) D(x, 1/k) = C(x/k, k).
(6.c) S(x, 1/k) = k * T(x/k, k).
(6.d) T(x, 1/k) = k * S(x/k, k).
(6.e) D(x, k) = C(k*x, 1/k).
(6.f) C(x, k) = D(k*x, 1/k).
(6.g) T(x, k) = (1/k) * S(k*x, 1/k).
(6.h) S(x, k) = (1/k) * T(k*x, 1/k).
Integrals.
(7.a) C = 1 + Integral S*D + x*S*D' dx.
(7.b) S = Integral C*D + x*C*D' dx.
(7.c) D = 1 + k^2 * Integral T*C + x*T*C' dx.
(7.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(8.a) C*C' = S*S'.
(8.b) D*D' = k^2*T*T'.
(9.a) C' = S * (D + x*D').
(9.b) S' = C * (D + x*D').
(9.c) D' = k^2 * T * (C + x*C').
(9.d) T' = D * (C + x*C').
(10.a) C' = S * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.b) S' = C * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.c) D' = k^2 * T * (C + x*S*D) / (1 - k^2*x^2*S*T).
(10.d) T' = D * (C + x*S*D) / (1 - k^2*x^2*S*T).
(11.a) (C + x*C') = (C + x*S*D) / (1 - k^2*x^2*S*T).
(11.b) (D + x*D') = (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
Logarithms.
(12.a) D = log(C + sqrt(C^2 - 1)) / x.
(12.b) C = log(D + sqrt(D^2 - 1)) / (k*x).
(12.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (k*x).
(12.d) S = sqrt(log(k*T + sqrt(1 + k^2*T^2))^2 - k^2*x^2) / (k*x).

A370432 Expansion of e.g.f. D(x,k) satisfying D(x,k) = cosh( k*x*cosh(x*D(x,k)) ), as a triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 12, 1, 0, 120, 420, 1, 0, 896, 36400, 10248, 1, 0, 5760, 2170560, 4858560, 196920, 1, 0, 33792, 102960000, 1127738304, 461126160, 3247860, 1, 0, 186368, 4083183104, 187282263168, 340884800256, 35248293080, 48361404, 1, 0, 983040, 141360128000, 25081621813248, 153279541958400, 76526954183680, 2290777550880, 669616080, 1
Offset: 0

Views

Author

Paul D. Hanna, Feb 19 2024

Keywords

Comments

The row sums equal A143601, the number of labeled odd degree trees with 2n+1 nodes.
Unsigned version of triangle A370332.
A row reversal of triangle A370430.

Examples

			E.g.f.: D(x,k) = 1 + (k^2)*x^2/2! + (12*k^2 + k^4)*x^4/4! + (120*k^2 + 420*k^4 + k^6)*x^6/6! + (896*k^2 + 36400*k^4 + 10248*k^6 + k^8)*x^8/8! + (5760*k^2 + 2170560*k^4 + 4858560*k^6 + 196920*k^8 + k^10)*x^10/10! + (33792*k^2 + 102960000*k^4 + 1127738304*k^6 + 461126160*k^8 + 3247860*k^10 + k^12)*x^12/12! + (186368*k^2 + 4083183104*k^4 + 187282263168*k^6 + 340884800256*k^8 + 35248293080*k^10 + 48361404*k^12 + k^14)*x^14/14! + ...
where D(x,k) = cosh( k*x*cosh(x*D(x,k)) ).
This triangle of coefficients a(n,j) of x^(2*n)*k^(2*j)/(2*n)! in D(x,k) begins
 1;
 0, 1;
 0, 12, 1;
 0, 120, 420, 1;
 0, 896, 36400, 10248, 1;
 0, 5760, 2170560, 4858560, 196920, 1;
 0, 33792, 102960000, 1127738304, 461126160, 3247860, 1;
 0, 186368, 4083183104, 187282263168, 340884800256, 35248293080, 48361404, 1;
 0, 983040, 141360128000, 25081621813248, 153279541958400, 76526954183680, 2290777550880, 669616080, 1; ...

Links

Crossrefs

Cf. A370430 (C), A370431 (S), A370433 (T), A143601 (row sums).
Cf. A370332.

Programs

  • PARI
    {a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cosh( x*cosh(k*x*C +Ox) );
S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
D = cosh( k*x*cosh(x*D +Ox));
T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))););
(2*n)! * polcoeff(polcoeff(D, 2*n, x), 2*j, k)}
for(n=0, 10, for(j=0, n, print1( a(n, j), ", ")); print(""))

Formula

E.g.f.: D(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n)*k^(2*j)/(2*n)! along with the related functions C = C(x,k), S = S(x,k), D = D(x,k), and T = T(x,k) satisfy the following formulas.
Definition.
(1.a) (C + S) = exp(x*D).
(1.b) (D + k*T) = exp(k*x*C).
(2.a) C^2 - S^2 = 1.
(2.b) D^2 - k^2*T^2 = 1.
Hyperbolic functions.
(3.a) C = cosh(x*D).
(3.b) S = sinh(x*D).
(3.c) D = cosh(k*x*C).
(3.d) T = (1/k) * sinh(k*x*C).
(4.a) C = cosh( x*cosh(k*x*C) ).
(4.b) S = sinh( x*cosh(k*x*sqrt(1 + S^2)) ).
(4.c) D = cosh( k*x*cosh(x*D) ).
(4.d) T = (1/k) * sinh( k*x*cosh(x*sqrt(1 + k^2*T^2)) ).
(5.a) (C*D + k*S*T) = cosh(x*D + k*x*C).
(5.b) (S*D + k*C*T) = sinh(x*D + k*x*C).
Transformations.
(6.a) C(x, 1/k) = D(x/k, k).
(6.b) D(x, 1/k) = C(x/k, k).
(6.c) S(x, 1/k) = k * T(x/k, k).
(6.d) T(x, 1/k) = k * S(x/k, k).
(6.e) D(x, k) = C(k*x, 1/k).
(6.f) C(x, k) = D(k*x, 1/k).
(6.g) T(x, k) = (1/k) * S(k*x, 1/k).
(6.h) S(x, k) = (1/k) * T(k*x, 1/k).
Integrals.
(7.a) C = 1 + Integral S*D + x*S*D' dx.
(7.b) S = Integral C*D + x*C*D' dx.
(7.c) D = 1 + k^2 * Integral T*C + x*T*C' dx.
(7.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(8.a) C*C' = S*S'.
(8.b) D*D' = k^2*T*T'.
(9.a) C' = S * (D + x*D').
(9.b) S' = C * (D + x*D').
(9.c) D' = k^2 * T * (C + x*C').
(9.d) T' = D * (C + x*C').
(10.a) C' = S * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.b) S' = C * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.c) D' = k^2 * T * (C + x*S*D) / (1 - k^2*x^2*S*T).
(10.d) T' = D * (C + x*S*D) / (1 - k^2*x^2*S*T).
(11.a) (C + x*C') = (C + x*S*D) / (1 - k^2*x^2*S*T).
(11.b) (D + x*D') = (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
Logarithms.
(12.a) D = log(C + sqrt(C^2 - 1)) / x.
(12.b) C = log(D + sqrt(D^2 - 1)) / (k*x).
(12.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (k*x).
(12.d) S = sqrt(log(k*T + sqrt(1 + k^2*T^2))^2 - k^2*x^2) / (k*x).

A143600 E.g.f. satisfies: A(x) = exp(x*A(x)/A(-x)).

Original entry on oeis.org

1, 1, 5, 25, 249, 2561, 40573, 641817, 14110001, 302279617, 8530496181, 230851019609, 7964867290537, 260618470319169, 10635790073585069, 408342804482252761, 19246730825243728737, 848289638051491455617, 45356940470607637151845, 2257054105205570995111833
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2008

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! +...
A LambertW identity yields the series:
A(x) = 1 + x/A(-x) + 3^1*x^2/2!/A(-x)^2 + 4^2*x^3/3!/A(-x)^3 + 5^3*x^4/4!/A(-x)^4 + 6^4*x^5/5!/A(-x)^5 +...+ (n+1)^(n-1)*x^n/n!/A(-x)^n +...
RELATED EXPANSIONS.
A(x)/A(-x) = F(2x) where F(x) is the e.g.f. of A058014:
A(x)/A(-x) = 1 + 2*x + 4*x^2/2! + 32*x^3/3! + 208*x^4/4! + 3072*x^5/5! +...
F(x) = 1 + x + 1*x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +...
which satisfies: F(x) = exp(x*(F(x) + 1/F(x))/2).
(A(x)/A(-x) + A(-x)/A(x))/2 = G(2x) where G(x) is the e.g.f. of A143601:
(A(x)/A(-x) + A(-x)/A(x))/2 = 1 + 4*x^2/2! + 208*x^4/4! + 34624*x^6/6! +...
G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
which satisfies G(x) = cosh(x*G(x)).

Links

Crossrefs

Cf. A058014, A143601, A007106.

Programs

  • PARI
    a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x*A/subst(A,x,-x)));n!*polcoeff(A,n)
  • PARI
    /* Formula Using a LambertW Identity: */
{a(n)=local(A=1);for(i=1,n,A=sum(k=0,n,(k+1)^(k-1)*x^k/k!/subst(A,x,-x)^k+x*O(x^n)));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Nov 05 2012

Formula

E.g.f. A(x) satisfies:
(1) A(x) = exp(x*exp(2x*G(2x))) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.
(2) [A(x)/A(-x) + A(-x)/A(x)]/2 = G(2x) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.
(3) A(x)/A(-x) = exp(x*[A(x)/A(-x) + A(-x)/A(x)]) = F(2x) where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.
(4) A(x) = Sum_{n>=0} (n+1)^(n-1) * x^n/n! / A(-x)^n.
(5) A(x)^m = Sum_{n>=0} m*(n+m)^(n-1) * x^n/n! / A(-x)^n.
(6) log(A(x)) = Sum_{n>=1} n^(n-1) * x^n/n! / A(-x)^n = x*A(x)/A(-x).
Formulas (4), (5), and (6) are due to LambertW identities. - Paul D. Hanna, Nov 05 2012
a(n) ~ c * n! / (n^(3/2) * r^n), where r = 0.33137170967459079... is the root of the equation sqrt(1+4*r^2) = log((1+sqrt(1+4*r^2))/(2*r)), and c = 1.35397895306096963692514418... if n is even, and c = 1.281887793570420328585518150... if n is odd. - Vaclav Kotesovec, Feb 25 2014

A370330 Expansion of e.g.f. C(x,k) satisfying C(x,k) = cos( x*cos(k*x*C(x,k)) ), as a triangle read by rows.

Original entry on oeis.org

1, -1, 0, 1, 12, 0, -1, -420, -120, 0, 1, 10248, 36400, 896, 0, -1, -196920, -4858560, -2170560, -5760, 0, 1, 3247860, 461126160, 1127738304, 102960000, 33792, 0, -1, -48361404, -35248293080, -340884800256, -187282263168, -4083183104, -186368, 0, 1, 669616080, 2290777550880, 76526954183680, 153279541958400, 25081621813248, 141360128000, 983040, 0
Offset: 0

Views

Author

Paul D. Hanna, Feb 19 2024

Keywords

Comments

The unsigned row sums equal A143601.
Signed version of triangle A370430.
A row reversal of triangle A370332.

Examples

			E.g.f.: C(x,k) = 1 - (1)*x^2/2! + (1 + 12*k^2)*x^4/4! - (1 + 420*k^2 + 120*k^4)*x^6/6! + (1 + 10248*k^2 + 36400*k^4 + 896*k^6)*x^8/8! - (1 + 196920*k^2 + 4858560*k^4 + 2170560*k^6 + 5760*k^8)*x^10/10! + (1 + 3247860*k^2 + 461126160*k^4 + 1127738304*k^6 + 102960000*k^8 + 33792*k^10)*x^12/12! - (1 + 48361404*k^2 + 35248293080*k^4 + 340884800256*k^6 + 187282263168*k^8 + 4083183104*k^10 + 186368*k^12)*x^14/14! + ...
where C(x,k) = cos( x*cos(k*x*C(x,k)) ).
This triangle of coefficients a(n,j) of x^(2*n)*k^(2*j)/(2*n)! in C(x,k) begins
  1;
 -1, 0;
 1, 12, 0;
 -1, -420, -120, 0;
 1, 10248, 36400, 896, 0;
 -1, -196920, -4858560, -2170560, -5760, 0;
 1, 3247860, 461126160, 1127738304, 102960000, 33792, 0;
 -1, -48361404, -35248293080, -340884800256, -187282263168, -4083183104, -186368, 0;
 1, 669616080, 2290777550880, 76526954183680, 153279541958400, 25081621813248, 141360128000, 983040, 0; ...

Crossrefs

Cf. A370331 (S), A370332 (D), A370333 (T).
Cf. A370430.

Programs

  • PARI
    {a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cos( x*cos(k*x*C +Ox) );
S = sin( x*cos(k*x*sqrt(1 - S^2 +Ox)) );
D = cos( k*x*cos(x*D +Ox));
T = (1/k)*sin( k*x*cos(x*sqrt(1 - k^2*T^2 +Ox))););
(2*n)! * polcoeff(polcoeff(C, 2*n, x), 2*j, k)}
for(n=0, 10, for(j=0, n, print1( a(n, j), ", ")); print(""))

Formula

E.g.f.: C(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n)*k^(2*j)/(2*n)! along with the related functions C = C(x,k), S = S(x,k), D = D(x,k), and T = T(x,k) satisfy the following formulas.
Definition.
(1.a) (C + i*S) = exp(i*x*D).
(1.b) (D + i*k*T) = exp(i*k*x*C).
(2.a) C^2 + S^2 = 1.
(2.b) D^2 + k^2*T^2 = 1.
Circular functions.
(3.a) C = cos(x*D).
(3.b) S = sin(x*D).
(3.c) D = cos(k*x*C).
(3.d) T = (1/k) * sin(k*x*C).
(4.a) C = cos( x*cos(k*x*C) ).
(4.b) S = sin( x*cos(k*x*sqrt(1 - S^2)) ).
(4.c) D = cos( k*x*cos(x*D) ).
(4.d) T = (1/k) * sin( k*x*cos(x*sqrt(1 - k^2*T^2)) ).
(5.a) (C*D - k*S*T) = cos(x*D + k*x*C).
(5.b) (S*D + k*C*T) = sin(x*D + k*x*C).
Transformations.
(6.a) C(x, 1/k) = D(x/k, k).
(6.b) D(x, 1/k) = C(x/k, k).
(6.c) S(x, 1/k) = k * T(x/k, k).
(6.d) T(x, 1/k) = k * S(x/k, k).
(6.e) D(x, k) = C(k*x, 1/k).
(6.f) C(x, k) = D(k*x, 1/k).
(6.g) T(x, k) = (1/k) * S(k*x, 1/k).
(6.h) S(x, k) = (1/k) * T(k*x, 1/k).
Integrals.
(7.a) C = 1 - Integral S*D + x*S*D' dx.
(7.b) S = Integral C*D + x*C*D' dx.
(7.c) D = 1 - k^2 * Integral T*C + x*T*C' dx.
(7.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(8.a) C*C' = -S*S'.
(8.b) D*D' = -k^2*T*T'.
(9.a) C' = -S * (D + x*D').
(9.b) S' = C * (D + x*D').
(9.c) D' = -k^2 * T * (C + x*C').
(9.d) T' = D * (C + x*C').
(10.a) C' = -S * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.b) S' = C * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.c) D' = -k^2 * T * (C - x*S*D) / (1 - k^2*x^2*S*T).
(10.d) T' = D * (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.a) (C + x*C') = (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.b) (D + x*D') = (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).

A370332 Expansion of e.g.f. D(x,k) satisfying D(x,k) = cos( k*x*cos(x*D(x,k)) ), as a triangle read by rows.

Original entry on oeis.org

1, 0, -1, 0, 12, 1, 0, -120, -420, -1, 0, 896, 36400, 10248, 1, 0, -5760, -2170560, -4858560, -196920, -1, 0, 33792, 102960000, 1127738304, 461126160, 3247860, 1, 0, -186368, -4083183104, -187282263168, -340884800256, -35248293080, -48361404, -1, 0, 983040, 141360128000, 25081621813248, 153279541958400, 76526954183680, 2290777550880, 669616080, 1
Offset: 0

Views

Author

Paul D. Hanna, Feb 19 2024

Keywords

Comments

The unsigned row sums equal A143601.
Signed version of triangle A370432.
A row reversal of triangle A370330.

Examples

			E.g.f.: D(x,k) = 1 - (k^2)*x^2/2! + (12*k^2 + k^4)*x^4/4! - (120*k^2 + 420*k^4 + k^6)*x^6/6! + (896*k^2 + 36400*k^4 + 10248*k^6 + k^8)*x^8/8! - (5760*k^2 + 2170560*k^4 + 4858560*k^6 + 196920*k^8 + k^10)*x^10/10! + (33792*k^2 + 102960000*k^4 + 1127738304*k^6 + 461126160*k^8 + 3247860*k^10 + k^12)*x^12/12! - (186368*k^2 + 4083183104*k^4 + 187282263168*k^6 + 340884800256*k^8 + 35248293080*k^10 + 48361404*k^12 + k^14)*x^14/14! + ...
where D(x,k) = cos( k*x*cos(x*D(x,k)) ).
This triangle of coefficients a(n,j) of x^(2*n)*k^(2*j)/(2*n)! in D(x,k) begins
 1;
 0, -1;
 0, 12, 1;
 0, -120, -420, -1;
 0, 896, 36400, 10248, 1;
 0, -5760, -2170560, -4858560, -196920, -1;
 0, 33792, 102960000, 1127738304, 461126160, 3247860, 1;
 0, -186368, -4083183104, -187282263168, -340884800256, -35248293080, -48361404, -1;
 0, 983040, 141360128000, 25081621813248, 153279541958400, 76526954183680, 2290777550880, 669616080, 1; ...

Crossrefs

Cf. A370330 (C), A370331 (S), A370333 (T).
Cf. A370432.

Programs

  • PARI
    {a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cos( x*cos(k*x*C +Ox) );
S = sin( x*cos(k*x*sqrt(1 - S^2 +Ox)) );
D = cos( k*x*cos(x*D +Ox));
T = (1/k)*sin( k*x*cos(x*sqrt(1 - k^2*T^2 +Ox))););
(2*n)! * polcoeff(polcoeff(D, 2*n, x), 2*j, k)}
for(n=0, 10, for(j=0, n, print1( a(n, j), ", ")); print(""))

Formula

E.g.f.: D(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n)*k^(2*j)/(2*n)! along with the related functions C = C(x,k), S = S(x,k), D = D(x,k), and T = T(x,k) satisfy the following formulas.
Definition.
(1.a) (C + i*S) = exp(i*x*D).
(1.b) (D + i*k*T) = exp(i*k*x*C).
(2.a) C^2 + S^2 = 1.
(2.b) D^2 + k^2*T^2 = 1.
Circular functions.
(3.a) C = cos(x*D).
(3.b) S = sin(x*D).
(3.c) D = cos(k*x*C).
(3.d) T = (1/k) * sin(k*x*C).
(4.a) C = cos( x*cos(k*x*C) ).
(4.b) S = sin( x*cos(k*x*sqrt(1 - S^2)) ).
(4.c) D = cos( k*x*cos(x*D) ).
(4.d) T = (1/k) * sin( k*x*cos(x*sqrt(1 - k^2*T^2)) ).
(5.a) (C*D - k*S*T) = cos(x*D + k*x*C).
(5.b) (S*D + k*C*T) = sin(x*D + k*x*C).
Transformations.
(6.a) C(x, 1/k) = D(x/k, k).
(6.b) D(x, 1/k) = C(x/k, k).
(6.c) S(x, 1/k) = k * T(x/k, k).
(6.d) T(x, 1/k) = k * S(x/k, k).
(6.e) D(x, k) = C(k*x, 1/k).
(6.f) C(x, k) = D(k*x, 1/k).
(6.g) T(x, k) = (1/k) * S(k*x, 1/k).
(6.h) S(x, k) = (1/k) * T(k*x, 1/k).
Integrals.
(7.a) C = 1 - Integral S*D + x*S*D' dx.
(7.b) S = Integral C*D + x*C*D' dx.
(7.c) D = 1 - k^2 * Integral T*C + x*T*C' dx.
(7.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(8.a) C*C' = -S*S'.
(8.b) D*D' = -k^2*T*T'.
(9.a) C' = -S * (D + x*D').
(9.b) S' = C * (D + x*D').
(9.c) D' = -k^2 * T * (C + x*C').
(9.d) T' = D * (C + x*C').
(10.a) C' = -S * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.b) S' = C * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.c) D' = -k^2 * T * (C - x*S*D) / (1 - k^2*x^2*S*T).
(10.d) T' = D * (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.a) (C + x*C') = (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.b) (D + x*D') = (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).

A372811 Expansion of e.g.f. C(x) satisfying C(x) = cosh( x*cosh(2*x*C(x)) ), where a(n) is the coefficient of x^(2*n)/(2*n)! in C(x) for n >= 0.

Original entry on oeis.org

1, 1, 49, 3601, 680737, 218915041, 105958624465, 74506995584113, 70436550855565633, 86815671664245679297, 135090335333407225545841, 258969022695032433287216593, 599973069857987759584855153249, 1652347283935245955005795151113121, 5336236426918250608377414155884578577
Offset: 0

Views

Author

Paul D. Hanna, May 16 2024

Keywords

Examples

			E.g.f: C(x) = 1 + x^2/2! + 49*x^4/4! + 3601*x^6/6! + 680737*x^8/8! + 218915041*x^10/10! + 105958624465*x^12/12! + 74506995584113*x^14/14! + ...
where C(x) = cosh( x*cosh(2*x*C(x)) ).
RELATED SERIES.
Related functions S(x), D(x), and T(x) are described below.
S(x) = x + 13*x^3/3! + 441*x^5/5! + 68069*x^7/7! + 15591025*x^9/9! + 6212017725*x^11/11! + 3652639410473*x^13/13! + 2963960104898581*x^15/15! + ...
where S(x) = sqrt(C(x)^2 - 1)
and S(x) = sinh( x*cosh( 2*x*sqrt(1 + S(x)^2) ) ).
D(x) = 1 + 4*x^2/2! + 64*x^4/4! + 7264*x^6/6! + 1242112*x^8/8! + 396112384*x^10/10! + 195196856320*x^12/12! + 135610245824512*x^14/14! + ...
where D(x) = cosh( 2*x*C(x) )
and D(x) = cosh( 2*x*cosh(x*D(x)) ).
T(x) = x + 7*x^3/3! + 381*x^5/5! + 50051*x^7/7! + 11899705*x^9/9! + 4787171775*x^11/11! + 2800735142453*x^13/13! + 2286983798222779*x^15/15! + ...
where T(x) = (1/2) * sinh( 2*x*C(x) )
and T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ).
SPECIFIC VALUES.
C(1/3) = 1.09195477630888952286167165173829275218422327069159600...
C(1/4) = 1.04077887287196699205886551058236704630676681245696946...
C(1/5) = 1.02363722685574465853941118194990482596731360834136139...
C(1/6) = 1.01558260957952973250484327981285267794556192126351939...
C(1/10) = 1.00520934315195311495083109289140774006817550223233669...

Links

Crossrefs

Cf. A370430 (k = 2), A372812 (S(x)), A372813 (D(x)), A372814 (T(x)), A143601.

Programs

  • PARI
    /* From C(x) = cosh( x*cosh(2*x*C(x)) ) */
{a(n) = my(C=1); for(i=0,n, C=truncate(C); C = cosh( x*cosh(2*x*C + x*O(x^(2*i))) ));
(2*n)! * polcoeff(C, 2*n, x)}
for(n=0, 30, print1( a(n), ", "))
  • PARI
    /* From A370430 at k = 2 */
{a(n, k = 2) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cosh( x*cosh(k*x*C +Ox) );
S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
D = cosh( k*x*cosh(x*D +Ox));
T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))); );
(2*n)! * polcoeff(C, 2*n, x)}
for(n=0, 30, print1( a(n), ", "))

Formula

a(n) = Sum_{j=0..n} A370430(n,j) * 2^(2*j).
E.g.f.: C(x) = Sum_{n>=0} a(n) * x^(2*n)/(2*n)! along with related functions denoted by C = C(x), S = S(x), D = D(x), and T = T(x) satisfy the following formulas.
Definition.
(1.a) (C + S) = exp(x*D).
(1.b) (D + 2*T) = exp(2*x*C).
(2.a) C^2 - S^2 = 1.
(2.b) D^2 - 4*T^2 = 1.
Hyperbolic functions.
(3.a) C = cosh(x*D).
(3.b) S = sinh(x*D).
(3.c) D = cosh(2*x*C).
(3.d) T = (1/2) * sinh(2*x*C).
(4.a) C = cosh( x*cosh(2*x*C) ).
(4.b) S = sinh( x*cosh(2*x*sqrt(1 + S^2)) ).
(4.c) D = cosh( 2*x*cosh(x*D) ).
(4.d) T = (1/2) * sinh( 2*x*cosh(x*sqrt(1 + 4*T^2)) ).
(5.a) (C*D + 2*S*T) = cosh(x*D + 2*x*C).
(5.b) (S*D + 2*C*T) = sinh(x*D + 2*x*C).
Integrals.
(6.a) C = 1 + Integral S*D + x*S*D' dx.
(6.b) S = Integral C*D + x*C*D' dx.
(6.c) D = 1 + 4 * Integral T*C + x*T*C' dx.
(6.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(7.a) C*C' = S*S'.
(7.b) D*D' = 4*T*T'.
(8.a) C' = S * (D + x*D').
(8.b) S' = C * (D + x*D').
(8.c) D' = 4 * T * (C + x*C').
(8.d) T' = D * (C + x*C').
(9.a) C' = S * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
(9.b) S' = C * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
(9.c) D' = 4 * T * (C + x*S*D) / (1 - 4*x^2*S*T).
(9.d) T' = D * (C + x*S*D) / (1 - 4*x^2*S*T).
(10.a) (C + x*C') = (C + x*S*D) / (1 - 4*x^2*S*T).
(10.b) (D + x*D') = (D + 4*x*T*C) / (1 - 4*x^2*S*T).
Logarithms.
(11.a) D = log(C + sqrt(C^2 - 1)) / x.
(11.b) C = log(D + sqrt(D^2 - 1)) / (2*x).
(11.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (2*x).
(11.d) S = sqrt(log(2*T + sqrt(1 + 4*T^2))^2 - 4*x^2) / (2*x).
The radius of convergence r and C(r) satisfy r = arccosh(C(r)) / cosh(2*r*C(r)) and 1 = 2*r^2 * sinh(2*r*C(r)) * sinh( r*cosh(2*r*C(r)) ), where r = 0.458693345589772637742719473602361341151810356245785213... and C(r) = 1.56301189045436141892741676499724550339640443730107995...

A372813 Expansion of e.g.f. D(x) satisfying D(x) = cosh( 2*x*cosh(x*D(x)) ), where a(n) is the coefficient of x^(2*n)/(2*n)! in D(x) for n >= 0.

Original entry on oeis.org

1, 4, 64, 7264, 1242112, 396112384, 195196856320, 135610245824512, 128604645225791488, 158304763492800790528, 246175295718345884041216, 471837283882871579572436992, 1092672848842771034323176914944, 3008542003438261199300841957228544, 9713742135846618809223753670120701952
Offset: 0

Views

Author

Paul D. Hanna, May 16 2024

Keywords

Examples

			E.g.f: D(x) = 1 + 4*x^2/2! + 64*x^4/4! + 7264*x^6/6! + 1242112*x^8/8! + 396112384*x^10/10! + 195196856320*x^12/12! + 135610245824512*x^14/14! + ...
and D(x) = cosh( 2*x*cosh(x*D(x)) ).
RELATED SERIES.
Related functions C(x), S(x), and T(x) are described below.
C(x) = 1 + x^2/2! + 49*x^4/4! + 3601*x^6/6! + 680737*x^8/8! + 218915041*x^10/10! + 105958624465*x^12/12! + 74506995584113*x^14/14! + ...
where C = cosh(x*D)
and C(x) = cosh( x*cosh(2*x*C(x)) ).
S(x) = x + 13*x^3/3! + 441*x^5/5! + 68069*x^7/7! + 15591025*x^9/9! + 6212017725*x^11/11! + 3652639410473*x^13/13! + 2963960104898581*x^15/15! + ...
where S(x) = S = sinh(x*D)
and S(x) = sinh( x*cosh( 2*x*sqrt(1 + S(x)^2) ) ).
T(x) = x + 7*x^3/3! + 381*x^5/5! + 50051*x^7/7! + 11899705*x^9/9! + 4787171775*x^11/11! + 2800735142453*x^13/13! + 2286983798222779*x^15/15! + ...
where T(x) = (1/2) * sqrt(D^2 - 1)
and T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ).
SPECIFIC VALUES.
D(1/3) = 1.276880244449228122993163054974488376796865611992370031...
D(1/4) = 1.138485942600540714616500323386982626365733417421170976...
D(1/5) = 1.085004369634098854421041251800873218914671999144038407...
D(1/6) = 1.057849764714936388260012199112395774792001649565003101...
D(1/10) = 1.020277074958546717842943931766605150247847706664020751...

Links

Crossrefs

Cf. A370432 (k = 2), A372811 (C(x)), A372812 (S(x)), A372814 (T(x)), A143601.

Programs

  • PARI
    /* From D(x) = cosh( 2*x*cosh(x*D(x)) ) */
{a(n) = my(D=1); for(i=0,n, D=truncate(D); D = cosh( 2*x*cosh(x*D + x*O(x^(2*i))) ));
(2*n)! * polcoeff(D, 2*n, x)}
for(n=0, 30, print1( a(n), ", "))
  • PARI
    /* From A370432 at k = 2 */
{a(n, k = 2) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cosh( x*cosh(k*x*C +Ox) );
S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
D = cosh( k*x*cosh(x*D +Ox));
T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))); );
(2*n)! * polcoeff(D, 2*n, x)}
for(n=0, 30, print1( a(n), ", "))

Formula

a(n) = Sum_{j=0..n} A370432(n,j) * 2^(2*j).
E.g.f.: D(x) = Sum_{n>=0} a(n) * x^(2*n)/(2*n)! along with related functions denoted by C = C(x), S = S(x), D = D(x), and T = T(x) satisfy the following formulas.
Definition.
(1.a) (C + S) = exp(x*D).
(1.b) (D + 2*T) = exp(2*x*C).
(2.a) C^2 - S^2 = 1.
(2.b) D^2 - 4*T^2 = 1.
Hyperbolic functions.
(3.a) C = cosh(x*D).
(3.b) S = sinh(x*D).
(3.c) D = cosh(2*x*C).
(3.d) T = (1/2) * sinh(2*x*C).
(4.a) C = cosh( x*cosh(2*x*C) ).
(4.b) S = sinh( x*cosh(2*x*sqrt(1 + S^2)) ).
(4.c) D = cosh( 2*x*cosh(x*D) ).
(4.d) T = (1/2) * sinh( 2*x*cosh(x*sqrt(1 + 4*T^2)) ).
(5.a) (C*D + 2*S*T) = cosh(x*D + 2*x*C).
(5.b) (S*D + 2*C*T) = sinh(x*D + 2*x*C).
Integrals.
(6.a) C = 1 + Integral S*D + x*S*D' dx.
(6.b) S = Integral C*D + x*C*D' dx.
(6.c) D = 1 + 4 * Integral T*C + x*T*C' dx.
(6.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(7.a) C*C' = S*S'.
(7.b) D*D' = 4*T*T'.
(8.a) C' = S * (D + x*D').
(8.b) S' = C * (D + x*D').
(8.c) D' = 4 * T * (C + x*C').
(8.d) T' = D * (C + x*C').
(9.a) C' = S * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
(9.b) S' = C * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
(9.c) D' = 4 * T * (C + x*S*D) / (1 - 4*x^2*S*T).
(9.d) T' = D * (C + x*S*D) / (1 - 4*x^2*S*T).
(10.a) (C + x*C') = (C + x*S*D) / (1 - 4*x^2*S*T).
(10.b) (D + x*D') = (D + 4*x*T*C) / (1 - 4*x^2*S*T).
Logarithms.
(11.a) D = log(C + sqrt(C^2 - 1)) / x.
(11.b) C = log(D + sqrt(D^2 - 1)) / (2*x).
(11.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (2*x).
(11.d) S = sqrt(log(2*T + sqrt(1 + 4*T^2))^2 - 4*x^2) / (2*x).
The radius of convergence r of e.g.f. D(x) is r = 0.458693345589772637742719473602361341151810356245785213... where D(r) = 2.216675597008249888019540624981069492182564304724769248...
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