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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.

%I A370432 #20 Feb 21 2024 10:22:31
%S A370432 1,0,1,0,12,1,0,120,420,1,0,896,36400,10248,1,0,5760,2170560,4858560,
%T A370432 196920,1,0,33792,102960000,1127738304,461126160,3247860,1,0,186368,
%U A370432 4083183104,187282263168,340884800256,35248293080,48361404,1,0,983040,141360128000,25081621813248,153279541958400,76526954183680,2290777550880,669616080,1
%N 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.
%C A370432 The row sums equal A143601, the number of labeled odd degree trees with 2n+1 nodes.
%C A370432 Unsigned version of triangle A370332.
%C A370432 A row reversal of triangle A370430.
%H A370432 Paul D. Hanna, <a href="/A370432/b370432.txt">Table of n, a(n) for n = 0..1325</a>
%F A370432 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.
%F A370432 Definition.
%F A370432 (1.a) (C + S) = exp(x*D).
%F A370432 (1.b) (D + k*T) = exp(k*x*C).
%F A370432 (2.a) C^2 - S^2 = 1.
%F A370432 (2.b) D^2 - k^2*T^2 = 1.
%F A370432 Hyperbolic functions.
%F A370432 (3.a) C = cosh(x*D).
%F A370432 (3.b) S = sinh(x*D).
%F A370432 (3.c) D = cosh(k*x*C).
%F A370432 (3.d) T = (1/k) * sinh(k*x*C).
%F A370432 (4.a) C = cosh( x*cosh(k*x*C) ).
%F A370432 (4.b) S = sinh( x*cosh(k*x*sqrt(1 + S^2)) ).
%F A370432 (4.c) D = cosh( k*x*cosh(x*D) ).
%F A370432 (4.d) T = (1/k) * sinh( k*x*cosh(x*sqrt(1 + k^2*T^2)) ).
%F A370432 (5.a) (C*D + k*S*T) = cosh(x*D + k*x*C).
%F A370432 (5.b) (S*D + k*C*T) = sinh(x*D + k*x*C).
%F A370432 Transformations.
%F A370432 (6.a) C(x, 1/k) = D(x/k, k).
%F A370432 (6.b) D(x, 1/k) = C(x/k, k).
%F A370432 (6.c) S(x, 1/k) = k * T(x/k, k).
%F A370432 (6.d) T(x, 1/k) = k * S(x/k, k).
%F A370432 (6.e) D(x, k) = C(k*x, 1/k).
%F A370432 (6.f) C(x, k) = D(k*x, 1/k).
%F A370432 (6.g) T(x, k) = (1/k) * S(k*x, 1/k).
%F A370432 (6.h) S(x, k) = (1/k) * T(k*x, 1/k).
%F A370432 Integrals.
%F A370432 (7.a) C = 1 + Integral S*D + x*S*D' dx.
%F A370432 (7.b) S = Integral C*D + x*C*D' dx.
%F A370432 (7.c) D = 1 + k^2 * Integral T*C + x*T*C' dx.
%F A370432 (7.d) T = Integral D*C + x*D*C' dx.
%F A370432 Derivatives (d/dx).
%F A370432 (8.a) C*C' = S*S'.
%F A370432 (8.b) D*D' = k^2*T*T'.
%F A370432 (9.a) C' = S * (D + x*D').
%F A370432 (9.b) S' = C * (D + x*D').
%F A370432 (9.c) D' = k^2 * T * (C + x*C').
%F A370432 (9.d) T' = D * (C + x*C').
%F A370432 (10.a) C' = S * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
%F A370432 (10.b) S' = C * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
%F A370432 (10.c) D' = k^2 * T * (C + x*S*D) / (1 - k^2*x^2*S*T).
%F A370432 (10.d) T' = D * (C + x*S*D) / (1 - k^2*x^2*S*T).
%F A370432 (11.a) (C + x*C') = (C + x*S*D) / (1 - k^2*x^2*S*T).
%F A370432 (11.b) (D + x*D') = (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
%F A370432 Logarithms.
%F A370432 (12.a) D = log(C + sqrt(C^2 - 1)) / x.
%F A370432 (12.b) C = log(D + sqrt(D^2 - 1)) / (k*x).
%F A370432 (12.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (k*x).
%F A370432 (12.d) S = sqrt(log(k*T + sqrt(1 + k^2*T^2))^2 - k^2*x^2) / (k*x).
%e A370432 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! + ...
%e A370432 where D(x,k) = cosh( k*x*cosh(x*D(x,k)) ).
%e A370432 This triangle of coefficients a(n,j) of x^(2*n)*k^(2*j)/(2*n)! in D(x,k) begins
%e A370432  1;
%e A370432  0, 1;
%e A370432  0, 12, 1;
%e A370432  0, 120, 420, 1;
%e A370432  0, 896, 36400, 10248, 1;
%e A370432  0, 5760, 2170560, 4858560, 196920, 1;
%e A370432  0, 33792, 102960000, 1127738304, 461126160, 3247860, 1;
%e A370432  0, 186368, 4083183104, 187282263168, 340884800256, 35248293080, 48361404, 1;
%e A370432  0, 983040, 141360128000, 25081621813248, 153279541958400, 76526954183680, 2290777550880, 669616080, 1; ...
%o A370432 (PARI) {a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
%o A370432 for(i=1, 2*n,
%o A370432 C = cosh( x*cosh(k*x*C +Ox) );
%o A370432 S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
%o A370432 D = cosh( k*x*cosh(x*D +Ox));
%o A370432 T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))););
%o A370432 (2*n)! * polcoeff(polcoeff(D, 2*n, x), 2*j, k)}
%o A370432 for(n=0, 10, for(j=0, n, print1( a(n, j), ", ")); print(""))
%Y A370432 Cf. A370430 (C), A370431 (S), A370433 (T), A143601 (row sums).
%Y A370432 Cf. A370332.
%K A370432 nonn,tabl
%O A370432 0,5
%A A370432 _Paul D. Hanna_, Feb 19 2024