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A357230
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a(n) = coefficient of x^(2*n-1)/(2*n-1)! in the expansion of the odd function S(x) defined by: S(x) = Integral Product_{n>=1} C(n,x)^(2*n-1) dx, where C(n,x) = (1 + S(x)^(2*n))^(1/(2*n)) for n >= 1.
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4
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1, 1, 19, 1339, 126121, 22936441, 6074972299, 2211448022179, 1068596557553041, 664819086091727281, 515877228619611775939, 487979294159765398810699, 553450493012139154035025081, 740913321416698764680850005161, 1156005387497662040937215014248379, 2079652309814657123017240379855646259
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OFFSET
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1,3
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LINKS
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FORMULA
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Odd function S(x) = Sum_{n>=1} a(n) * x^(2*n-1)/(2*n-1)! and related even functions C(n,x), n >= 1, satisfy the following relations.
(1.a) C(n,x)^(2*n) - S(x)^(2*n) = 1, for n >= 1.
(1.b) C'(n,x) * C(n,x)^(2*n-1) = S'(x) * S(x)^(2*n-1), for n >= 1.
(2.a) S(x) = Integral Product_{n>=1} C(n,x)^(2*n-1) dx.
(2.b) S(x) = Integral 1 / Product_{n>=1} C(n,x) * (1 - S(x)^(4*n-2)) dx.
(2.c) C(n,x) = 1 + Integral S(x)^(2*n-1)/C(n,x)^(2*n-1) * Product_{k>=1} C(k,x)^(2*k-1) dx, for n >= 1.
(3) C(n,x)^n + S(x)^n = exp( n * Integral S'(x)*S(x)^(n-1) / C(n,x)^n dx ), for n >= 1.
(3.a) C(1,x) + S(x) = exp( Integral S'(x) / C(1,x) dx ).
(3.b) C(2,x)^2 + S(x)^2 = exp( 2 * Integral S'(x)*S(x) / C(2,x)^2 dx ).
(3.c) C(3,x)^3 + S(x)^3 = exp( 3 * Integral S'(x)*S(x)^2 / C(3,x)^3 dx ).
(3.d) C(4,x)^4 + S(x)^4 = exp( 4 * Integral S'(x)*S(x)^3 / C(4,x)^4 dx ).
(4.a) C(n,x)^n = cosh( n * Integral S'(x)*S(x)^(n-1) / C(n,x)^n dx ), for n >= 1.
(4.b) S(x)^n = sinh( n * Integral S'(x)*S(x)^(n-1) / C(n,x)^n dx ), for n >= 1.
(5) C(n,x)^(2*n) + S(x)^(2*n) = cosh( 2*n * Integral S'(x)*S(x)^(n-1) / C(n,x)^n dx ), for n >= 1.
(5.a) 2*C(n,x)^(2*n) = 1 + cosh( 2*n * Integral S'(x)*S(x)^(n-1) / C(n,x)^n dx ), for n >= 1.
(5.b) 2*S(x)^(2*n) = -1 + cosh( 2*n * Integral S'(x)*S(x)^(n-1) / C(n,x)^n dx ), for n >= 1.
(6.a) S(x) = Series_Reversion( Integral Product_{n>=1} (1 + x^(2*n))^(-(2*n-1)/(2*n)) dx ).
(6.b) S(x) = Series_Reversion( Integral Product_{n>=1} (1 + x^(2*n))^(1/(2*n)) * (1 - x^(4*n-2)) dx ).
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EXAMPLE
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E.g.f.: S(x) = x + x^3/3! + 19*x^5/5! + 1339*x^7/7! + 126121*x^9/9! + 22936441*x^11/11! + 6074972299*x^13/13! + 2211448022179*x^15/15! + 1068596557553041*x^17/17! + 664819086091727281*x^19/19! + ...
such that
S(x) = Series_Reversion( Integral 1/((1 + x^2)^(1/2) * (1 + x^4)^(3/4) * (1 + x^6)^(5/6) * (1 + x^8)^(7/8) * ... * (1 + x^(2*n))^((2*n-1)/(2*n)) * ...) dx )
also,
S(x) = Integral C(1,x) * C(2,x)^3 * C(3,x)^5 * C(4,x)^7 * ... * C(n,x)^(2*n-1) * ... dx,
where
C(1,x) = (1 + S(x)^2)^(1/2) = 1 + x^2/2! + x^4/4! + 109*x^6/6! + 8689*x^8/8! + 1053481*x^10/10! + 243813361*x^12/12! + 75186825109*x^14/14! + ...
C(2,x) = (1 + S(x)^4)^(1/4) = 1 + 6*x^4/4! + 120*x^6/6! + 4284*x^8/8! + 814560*x^10/10! + 188920776*x^12/12! + 57298681440*x^14/14! + ...
C(3,x) = (1 + S(x)^6)^(1/6) = 1 + 120*x^6/6! + 6720*x^8/8! + 826560*x^10/10! + 164588160*x^12/12! + 46958271360*x^14/14! + ...
C(4,x) = (1 + S(x)^8)^(1/8) = 1 + 5040*x^8/8! + 604800*x^10/10! + 122411520*x^12/12! + 42090048000*x^14/14! + ...
C(5,x) = (1 + S(x)^10)^(1/10) = 1 + 362880*x^10/10! + 79833600*x^12/12! + 24700515840*x^14/14! + ...
...
C(2,x)^3 = 1 + 18*x^4/4! + 360*x^6/6! + 20412*x^8/8! + 3350880*x^10/10! + 690504408*x^12/12! + 213236543520*x^14/14! + ...
C(3,x)^5 = 1 + 600*x^6/6! + 33600*x^8/8! + 4132800*x^10/10! + 955996800*x^12/12! + 283223740800*x^14/14! + ...
C(4,x)^7 = 1 + 35280*x^8/8! + 4233600*x^10/10! + 856880640*x^12/12! + 294630336000*x^14/14! + ...
C(5,x)^9 = 1 + 3265920*x^10/10! + 718502400*x^12/12! + 222304642560*x^14/14! + ...
...
Illustrate the expansions of log( C(n,x)^n + S(x)^n ) for some initial values of n:
log(C(1,x) + S(x)) = Integral S'(x) / C(1,x) dx = x + 18*x^5/5! + 960*x^7/7! + 89292*x^9/9! + 17251200*x^11/11! + 4474273848*x^13/13! + ...
log(C(2,x)^2 + S(x)^2) = 2 * Integral S'(x)*S(x) / C(2,x)^2 dx = 2*x^2/2! + 8*x^4/4! + 128*x^6/6! + 16832*x^8/8! + 2380352*x^10/10! + 508289408*x^12/12! + ...
log(C(3,x)^3 + S(x)^3) = 3 * Integral S'(x)*S(x)^2 / C(3,x)^3 dx = 6*x^3/3! + 60*x^5/5! + 2814*x^7/7! + 287880*x^9/9! + 45774366*x^11/11! + ...
log(C(4,x)^4 + S(x)^4) = 4 * Integral S'(x)*S(x)^3 / C(4,x)^4 dx = 24*x^4/4! + 480*x^6/6! + 32256*x^8/8! + 5072640*x^10/10! + 938302464*x^12/12! + ...
...
SERIES REVERSION.
Let R(x) be the series reversion of S(x) so that S(R(x)) = x, then
R(x) = Integral Product_{n>=1} 1/(1 + x^(2*n))^((2*n-1)/(2*n)) dx,
also
R(x) = Integral 1/Product_{n>=1} (1 + x^(2*n))^(1/(2*n)) * (1 - x^(4*n-2)) dx,
where
R(x) = x - x^3/3! - 9*x^5/5! - 555*x^7/7! + 7665*x^9/9! - 1777545*x^11/11! + 114147495*x^13/13! - 27004972995*x^15/15! + 20805419059425*x^17/17! - 4204053743915025*x^19/19! + ... + A357229(n)*x^(2*n-1)/(2*n-1)! + ...
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PROG
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(PARI) /* Using Series Reversion of Integral of infinite product */
{a(n) = my(S); S = serreverse( intformal( prod(k=1, n, (1 + x^(2*k) + O(x^(2*n)) )^(-(2*k-1)/(2*k)) ) ) ); (2*n-1)! * polcoeff(S, 2*n-1)}
for(n=1, 20, print1(a(n), ", "))
(PARI) /* Using Integral of infinite product */
{a(n) = my(S=x); for(i=1, n,
S = intformal( prod(k=1, n, (1 + S^(2*k) + O(x^(2*n)))^((2*k-1)/(2*k)) ) ) ); (2*n-1)! * polcoeff(S, 2*n-1)}
for(n=1, 20, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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