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.

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

Author

Paul D. Hanna, Feb 19 2024

Keywords

• nonn
• tabl

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

• Paul D. Hanna, Table of n, a(n) for n = 0..1325

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).