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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.

%I A370433 #17 Feb 21 2024 10:21:30
%S A370433 1,3,1,5,90,1,7,3675,2205,1,9,107604,532350,46116,1,11,2436885,
%T A370433 74042430,52887450,812295,1,13,46444398,7663602375,24609789204,
%U A370433 4257556875,12666654,1,15,785872815,643910782515,7510986678195,5841878527485,292686719325,181355265,1,17,12196578600,45911000082220,1766457334617976,4451226370197750,1124109212938712,17658076954700,2439315720,1
%N 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.
%C A370433 The row sums equal A007106, the number of labeled odd degree trees with 2n nodes.
%C A370433 Unsigned version of triangle A370333.
%C A370433 A row reversal of triangle A370431.
%H A370433 Paul D. Hanna, <a href="/A370433/b370433.txt">Table of n, a(n) for n = 0..1325</a>
%F A370433 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.
%F A370433 Definition.
%F A370433 (1.a) (C + S) = exp(x*D).
%F A370433 (1.b) (D + k*T) = exp(k*x*C).
%F A370433 (2.a) C^2 - S^2 = 1.
%F A370433 (2.b) D^2 - k^2*T^2 = 1.
%F A370433 Hyperbolic functions.
%F A370433 (3.a) C = cosh(x*D).
%F A370433 (3.b) S = sinh(x*D).
%F A370433 (3.c) D = cosh(k*x*C).
%F A370433 (3.d) T = (1/k) * sinh(k*x*C).
%F A370433 (4.a) C = cosh( x*cosh(k*x*C) ).
%F A370433 (4.b) S = sinh( x*cosh(k*x*sqrt(1 + S^2)) ).
%F A370433 (4.c) D = cosh( k*x*cosh(x*D) ).
%F A370433 (4.d) T = (1/k) * sinh( k*x*cosh(x*sqrt(1 + k^2*T^2)) ).
%F A370433 (5.a) (C*D + k*S*T) = cosh(x*D + k*x*C).
%F A370433 (5.b) (S*D + k*C*T) = sinh(x*D + k*x*C).
%F A370433 Transformations.
%F A370433 (6.a) C(x, 1/k) = D(x/k, k).
%F A370433 (6.b) D(x, 1/k) = C(x/k, k).
%F A370433 (6.c) S(x, 1/k) = k * T(x/k, k).
%F A370433 (6.d) T(x, 1/k) = k * S(x/k, k).
%F A370433 (6.e) D(x, k) = C(k*x, 1/k).
%F A370433 (6.f) C(x, k) = D(k*x, 1/k).
%F A370433 (6.g) T(x, k) = (1/k) * S(k*x, 1/k).
%F A370433 (6.h) S(x, k) = (1/k) * T(k*x, 1/k).
%F A370433 Integrals.
%F A370433 (7.a) C = 1 + Integral S*D + x*S*D' dx.
%F A370433 (7.b) S = Integral C*D + x*C*D' dx.
%F A370433 (7.c) D = 1 + k^2 * Integral T*C + x*T*C' dx.
%F A370433 (7.d) T = Integral D*C + x*D*C' dx.
%F A370433 Derivatives (d/dx).
%F A370433 (8.a) C*C' = S*S'.
%F A370433 (8.b) D*D' = k^2*T*T'.
%F A370433 (9.a) C' = S * (D + x*D').
%F A370433 (9.b) S' = C * (D + x*D').
%F A370433 (9.c) D' = k^2 * T * (C + x*C').
%F A370433 (9.d) T' = D * (C + x*C').
%F A370433 (10.a) C' = S * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
%F A370433 (10.b) S' = C * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
%F A370433 (10.c) D' = k^2 * T * (C + x*S*D) / (1 - k^2*x^2*S*T).
%F A370433 (10.d) T' = D * (C + x*S*D) / (1 - k^2*x^2*S*T).
%F A370433 (11.a) (C + x*C') = (C + x*S*D) / (1 - k^2*x^2*S*T).
%F A370433 (11.b) (D + x*D') = (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
%F A370433 Logarithms.
%F A370433 (12.a) D = log(C + sqrt(C^2 - 1)) / x.
%F A370433 (12.b) C = log(D + sqrt(D^2 - 1)) / (k*x).
%F A370433 (12.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (k*x).
%F A370433 (12.d) S = sqrt(log(k*T + sqrt(1 + k^2*T^2))^2 - k^2*x^2) / (k*x).
%e A370433 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! + ...
%e A370433 where T(x,k) = (1/k) * sinh( k*x*cosh(x*sqrt(1 + k^2*T(x,k)^2)) ).
%e A370433 This triangle of coefficients a(n,j) of x^(2*n+1)*k^(2*j)/(2*n+1)! in T(x,k) begins
%e A370433  1;
%e A370433  3, 1;
%e A370433  5, 90, 1;
%e A370433  7, 3675, 2205, 1;
%e A370433  9, 107604, 532350, 46116, 1;
%e A370433  11, 2436885, 74042430, 52887450, 812295, 1;
%e A370433  13, 46444398, 7663602375, 24609789204, 4257556875, 12666654, 1;
%e A370433  15, 785872815, 643910782515, 7510986678195, 5841878527485, 292686719325, 181355265, 1;
%e A370433  17, 12196578600, 45911000082220, 1766457334617976, 4451226370197750, 1124109212938712, 17658076954700, 2439315720, 1; ...
%o A370433 (PARI) {a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n+1)));
%o A370433 for(i=1, 2*n+1,
%o A370433 C = cosh( x*cosh(k*x*C +Ox) );
%o A370433 S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
%o A370433 D = cosh( k*x*cosh(x*D +Ox));
%o A370433 T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))););
%o A370433 (2*n+1)! *polcoeff(polcoeff(T, 2*n+1, x), 2*j, k)}
%o A370433 for(n=0, 10, for(k=0, n, print1( a(n, k), ", ")); print(""))
%Y A370433 Cf. A370430 (C), A370431 (S), A370432 (D), A007106 (row sums).
%Y A370433 Cf. A370333.
%K A370433 nonn,tabl
%O A370433 0,2
%A A370433 _Paul D. Hanna_, Feb 19 2024