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.

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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).

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).

A370333 Expansion of e.g.f. T(x,k) satisfying T(x,k) = (1/k) * sin( k*x*cos(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
Offset: 0

Views

Author

Paul D. Hanna, Feb 19 2024

Keywords

Comments

Unsigned row sums equal A007106.
Signed version of triangle A370433.
A row reversal of triangle A370331.

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) * sin( k*x*cos(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; ...

Crossrefs

Cf. A370330 (C), A370331 (S), A370332 (D).
Cf. A370433.

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 = 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+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 + 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).

A372814 Expansion of e.g.f. T(x) satisfying T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ), where a(n) is the coefficient of x^(2*n+1)/(2*n+1)! in T(x) for n >= 0.

Original entry on oeis.org

1, 7, 381, 50051, 11899705, 4787171775, 2800735142453, 2286983798222779, 2476757127978318705, 3434360322639603491447, 5940446259665147492879341, 12533181362722474751715110643, 31687559294370295303515685200041, 94578054008984518849163257005668911
Offset: 0

Views

Author

Paul D. Hanna, May 16 2024

Keywords

Examples

			E.g.f: 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! + ...
and T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ).
RELATED SERIES.
Related functions C(x), S(x), and D(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(x) = cosh( x*sqrt(1 + 4*T(x)^2) )
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) = sinh( x*sqrt(1 + 4*T(x)^2) )
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) = sqrt(1 + 4*T(x)^2)
and D(x) = cosh( 2*x*cosh(x*D(x)) ).
SPECIFIC VALUES.
T(1/3) = 0.396995956737823895057073833881324565170496359298875629...
T(1/4) = 0.272098438758403062037693234896743179247292418834110582...
T(1/5) = 0.210496129492378206205507948355083031094691532837329579...
T(1/6) = 0.172515307079869202329210004732936197729646250794708377...
T(1/10) = 0.101199443780546489307264980334019941975165460819500909...

Links

Crossrefs

Cf. A370433 (k = 2), A372811 (C(x)), A372812 (S(x)), A372813 (D(x)), A007106.

Programs

  • PARI
    /* From T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ) */
{a(n) = my(T=x); for(i=0,n, T=truncate(T); T = (1/2) * sinh( 2*x*cosh(x*sqrt(1 + 4*T^2 + x*O(x^(2*i)) )) ));
(2*n+1)! * polcoeff(T, 2*n+1, x)}
for(n=0, 30, print1( a(n), ", "))
  • PARI
    /* From A370433 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+1)! * polcoeff(T, 2*n+1, x)}
for(n=0, 30, print1( a(n), ", "))

Formula

a(n) = Sum_{j=0..n} A370433(n,j) * 2^(2*j).
E.g.f.: T(x) = Sum_{n>=0} a(n) * x^(2*n+1)/(2*n+1)! 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. T(x) is r = 0.458693345589772637742719473602361341151810356245785213... where T(r) = 0.989147448863398861152401518907145018698758995598697027...
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