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 1-6 of 6 results.

A143601 Number of labeled odd-degree trees with 2n+1 nodes.

Original entry on oeis.org

1, 1, 13, 541, 47545, 7231801, 1695106117, 567547087381, 257320926233329, 151856004814953841, 113144789723082206461, 103890621918675777804301, 115270544419577901796226473, 152049571406030636219959644841
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2008, May 27 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! + ...
The e.g.f. of A007106 (a bisection of A058014) is given by:
sqrt(A(x)^2 - 1) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! + ...
The e.g.f. of A058014 is given by:
F(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! + ...
where A(x) = [F(x) + F(-x)]/2 and exp(x*A(x)) = F(x).
The e.g.f. of A143600 is given by:
G(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! + ...
where A(2x) = [G(x)/G(-x) + G(-x)/G(x)]/2.

Links

Crossrefs

Cf. A058014, A143600, A007106, A309204.
Cf. A370430, A370432.

Programs

  • Mathematica
    Table[(2*n)!*CoefficientList[1/x*InverseSeries[Series[x/Cosh[x],{x,0,41}],x],x][[2*n+1]],{n,0,20}] (* Vaclav Kotesovec, Jan 10 2014 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=cosh(x*A));n!*polcoeff(A,n)}
  • PARI
    {a(n)=(2*n)!*polcoeff(cosh(x+x*O(x^(2*n)))^(2*n+1)/(2*n+1),2*n)} \\ Paul D. Hanna, Aug 29 2008
  • PARI
    {a(n) = sum(k=0,n, binomial(2*n+1,k) * (2*n+1-2*k)^(2*n) / ((2*n+1) * 2^(2*n)) )}
for(n=0,30, print1(a(n),", ")) \\ Paul D. Hanna, Feb 19 2024

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! satisfies the following formulas.
(1) A(x) = cosh(x*A(x)).
(2) A(x) = (1/x)*Series_Reversion( x/cosh(x) ).
(3) sqrt(A(x)^2 - 1) = e.g.f. of A007106.
(4) exp(x*A(x)) = A(x) + sqrt(A(x)^2-1) = e.g.f. of A058014.
(5) A(x) = [F(x) + F(-x)]/2 where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.
(6) A(2*x) = [G(x)/G(-x) + G(-x)/G(x)]/2 where G(x) = exp(x*G(x)/G(-x)) = e.g.f. of A143600.
From Paul D. Hanna, Aug 29 2008: (Start)
(7) A(x/cosh(x)) = cosh(x).
(8) a(n) = (2n)!*[x^(2n)] cosh(x)^(2n+1)/(2n+1). (End)
(9) a(n) = Sum_{k=0..n} binomial(2*n+1,k) * (2*n+1 - 2*k)^(2*n) / ((2*n+1) * 2^(2*n)). [See formula by Christophe Vignat in A309204.] - Paul D. Hanna, Feb 19 2024
a(n) ~ 2^(2*n) * n^(2*n-1) * (s^2-1)^(n+1/2) / exp(2*n), where s = 1.810170580698977274512829... is the root of the equation sqrt(s^2-1) * log(s + sqrt(s^2-1)) = s. - Vaclav Kotesovec, Jan 10 2014
Radius of convergence r = 0.66274341934918158097474... = 1/sqrt(s^2-1) and A(r) = s (given above) satisfy r = 1/sinh(r*A(r)) and A(r) = cosh(r*A(r)). - Paul D. Hanna, Mar 04 2024

Extensions

Edited by Paul D. Hanna, May 27 2009

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).

A370431 Expansion of e.g.f. S(x,k) satisfying S(x,k) = sinh( x*cosh(k*x*sqrt(1 + S(x,k)^2)) ), as a triangle read by rows.

Original entry on oeis.org

1, 1, 3, 1, 90, 5, 1, 2205, 3675, 7, 1, 46116, 532350, 107604, 9, 1, 812295, 52887450, 74042430, 2436885, 11, 1, 12666654, 4257556875, 24609789204, 7663602375, 46444398, 13, 1, 181355265, 292686719325, 5841878527485, 7510986678195, 643910782515, 785872815, 15, 1, 2439315720, 17658076954700, 1124109212938712, 4451226370197750, 1766457334617976, 45911000082220, 12196578600, 17
Offset: 0

Views

Author

Paul D. Hanna, Feb 19 2024

Keywords

Comments

The row sums equal A007106, the number of labeled odd degree trees with 2n nodes.
Unsigned version of triangle A370331.
A row reversal of triangle A370433.

Examples

			E.g.f.: S(x,k) = x + (1 + 3*k^2)*x^3/3! + (1 + 90*k^2 + 5*k^4)*x^5/5! + (1 + 2205*k^2 + 3675*k^4 + 7*k^6)*x^7/7! + (1 + 46116*k^2 + 532350*k^4 + 107604*k^6 + 9*k^8)*x^9/9! + (1 + 812295*k^2 + 52887450*k^4 + 74042430*k^6 + 2436885*k^8 + 11*k^10)*x^11/11! + (1 + 12666654*k^2 + 4257556875*k^4 + 24609789204*k^6 + 7663602375*k^8 + 46444398*k^10 + 13*k^12)*x^13/13! + ...
where S(x,k) = sinh( x*cosh(k*x*sqrt(1 + S(x,k)^2)) ).
This triangle of coefficients a(n,j) of x^(2*n+1)*k^(2*j)/(2*n+1)! in S(x,k) begins
 1;
 1, 3;
 1, 90, 5;
 1, 2205, 3675, 7;
 1, 46116, 532350, 107604, 9;
 1, 812295, 52887450, 74042430, 2436885, 11;
 1, 12666654, 4257556875, 24609789204, 7663602375, 46444398, 13;
 1, 181355265, 292686719325, 5841878527485, 7510986678195, 643910782515, 785872815, 15;
 1, 2439315720, 17658076954700, 1124109212938712, 4451226370197750, 1766457334617976, 45911000082220, 12196578600, 17;
 1, 31284206667, 957418671788100, 185331614609361948, 1972848836100689118, 2411259688567508922, 344187284274529332, 2872256015364300, 177277171113, 19; ...

Links

Crossrefs

Cf. A370430 (C), A370432 (D), A370433 (T), A007106 (row sums).
Cf. A370331.

Programs

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

Formula

E.g.f.: S(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n+1)*k^(2*j)/(2*n+1)! 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).

A370433 Expansion of e.g.f. T(x,k) satisfying T(x,k) = (1/k) * sinh( k*x*cosh(x*sqrt(1 + k^2*T(x,k)^2)) ), as a triangle read by rows.

Original entry on oeis.org

1, 3, 1, 5, 90, 1, 7, 3675, 2205, 1, 9, 107604, 532350, 46116, 1, 11, 2436885, 74042430, 52887450, 812295, 1, 13, 46444398, 7663602375, 24609789204, 4257556875, 12666654, 1, 15, 785872815, 643910782515, 7510986678195, 5841878527485, 292686719325, 181355265, 1, 17, 12196578600, 45911000082220, 1766457334617976, 4451226370197750, 1124109212938712, 17658076954700, 2439315720, 1
Offset: 0

Views

Author

Paul D. Hanna, Feb 19 2024

Keywords

Comments

The row sums equal A007106, the number of labeled odd degree trees with 2n nodes.
Unsigned version of triangle A370333.
A row reversal of triangle A370431.

Examples

			E.g.f.: T(x,k) = x + (3 + k^2)*x^3/3! + (5 + 90*k^2 + k^4)*x^5/5! + (7 + 3675*k^2 + 2205*k^4 + k^6)*x^7/7! + (9 + 107604*k^2 + 532350*k^4 + 46116*k^6 + k^8)*x^9/9! + (11 + 2436885*k^2 + 74042430*k^4 + 52887450*k^6 + 812295*k^8 + k^10)*x^11/11! + (13 + 46444398*k^2 + 7663602375*k^4 + 24609789204*k^6 + 4257556875*k^8 + 12666654*k^10 + k^12)*x^13/13! + ...
where T(x,k) = (1/k) * sinh( k*x*cosh(x*sqrt(1 + k^2*T(x,k)^2)) ).
This triangle of coefficients a(n,j) of x^(2*n+1)*k^(2*j)/(2*n+1)! in T(x,k) begins
 1;
 3, 1;
 5, 90, 1;
 7, 3675, 2205, 1;
 9, 107604, 532350, 46116, 1;
 11, 2436885, 74042430, 52887450, 812295, 1;
 13, 46444398, 7663602375, 24609789204, 4257556875, 12666654, 1;
 15, 785872815, 643910782515, 7510986678195, 5841878527485, 292686719325, 181355265, 1;
 17, 12196578600, 45911000082220, 1766457334617976, 4451226370197750, 1124109212938712, 17658076954700, 2439315720, 1; ...

Links

Crossrefs

Cf. A370430 (C), A370431 (S), A370432 (D), A007106 (row sums).
Cf. A370333.

Programs

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

Formula

E.g.f.: T(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n+1)*k^(2*j)/(2*n+1)! 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).

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).

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...
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