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 A325291 E.g.f. C(x), where C(x*y) + S(x*y) = exp( Integral Integral C(x*y) dx dy ) such that C(x)^2 - S(x)^2 = 1. 4
 1, 2, 56, 8336, 3985792, 4679517952, 11427218287616, 51793067942397952, 400951893341645930496, 4975999084909976839454720, 94178912073481319162642169856, 2610878440961060713599511173791744, 102545703927828194073741484514193965056, 5548919569628098800740786379865766154469376, 403949193167852851803947801218003477783686152192 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Unsigned version of A326551. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..200 FORMULA E.g.f. C(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!^2, where series C(x) and related series S(x) satisfy the following relations. (1.a) C(x)^2 - S(x)^2 = 1. (1.b) C'(x)/S(x) = S'(x)/C(x) = 1/x * Integral C(x) dx. (2.a) S(x) = Integral C(x)/x * (Integral C(x) dx) dx. (2.b) C(x) = 1 + Integral S(x)/x * (Integral C(x) dx) dx. (3.a) C(x) + S(x) = exp( Integral 1/x * (Integral C(x) dx) dx ). (3.b) C(x) = cosh( Integral 1/x * (Integral C(x) dx) dx ). (3.c) S(x) = sinh( Integral 1/x * (Integral C(x) dx) dx ). Integration. (4.a) S(x*y) = Integral C(x*y) * (Integral C(x*y) dy) dx. (4.b) C(x*y) = 1 + Integral S(x*y) * (Integral C(x*y) dy) dx. (4.c) S(x*y) = Integral C(x*y) * (Integral C(x*y) dx) dy. (4.d) C(x*y) = 1 + Integral S(x*y) * (Integral C(x*y) dx) dy. Exponential. (5.a) C(x*y) + S(x*y) = exp( Integral Integral C(x*y) dx dy ). (5.b) C(x*y) = cosh( Integral Integral C(x*y) dx dy ). (5.c) S(x*y) = sinh( Integral Integral C(x*y) dx dy ). Derivatives. (6.a) d/dx S(x*y) = C(x*y) * Integral C(x*y) dy. (6.b) d/dx C(x*y) = S(x*y) * Integral C(x*y) dy. (6.c) d/dy S(x*y) = C(x*y) * Integral C(x*y) dx. (6.d) d/dy C(x*y) = S(x*y) * Integral C(x*y) dx. EXAMPLE E.g.f. C(x) = 1 + 2*x^2/2!^2 + 56*x^4/4!^2 + 8336*x^6/6!^2 + 3985792*x^8/8!^2 + 4679517952*x^10/10!^2 + 11427218287616*x^12/12!^2 + 51793067942397952*x^14/14!^2 + 400951893341645930496*x^16/16!^2 + 4975999084909976839454720*x^18/18!^2 + 94178912073481319162642169856*x^20/20!^2 + ... where C(x) = cosh( Integral 1/x * (Integral C(x) dx) dx ), also, C(x*y) = cosh( Integral Integral C(x*y) dx dy ). RELATED SERIES. S(x) = x + 8*x^3/3!^2 + 576*x^5/5!^2 + 160768*x^7/7!^2 + 123535360*x^9/9!^2 + 212713734144*x^11/11!^2 + 716196297048064*x^13/13!^2 + 4280584942657732608*x^15/15!^2 + 42250703121584165486592*x^17/17!^2 + 651154631135458759089848320*x^19/19!^2 + 14983590319172065236171175755776*x^21/21!^2 + ... where S(x) = sinh( Integral 1/x * (Integral C(x) dx) dx ), also, S(x*y) = sinh( Integral Integral C(x*y) dx dy ). SPECIFIC VALUES. At x = 1/2, C(1/2) = 1.13133757946411922642102833324416139... S(1/2) = 0.52907912329606456055608764850290077... log(C(1/2) + S(1/2)) = 0.50706859662590456104854330721421537... At x = 1, C(1) = 1.61616724447561044622618032294959193... S(1) = 1.26964426597212165112687564431552303... log(C(1) + S(1)) = 1.05980614652360497313310791544203867... At x = 2, C(2) = 7.0181980831554020705059330009720760... S(2) = 6.9465894030384550946994132182413166... log(C(2) + S(2)) = 2.636538981679765615420983831302958... At x = 3, the power series for C(x) and S(x) diverge. PROG (PARI) {a(n) = my(C=1, S=x); for(i=1, 2*n, S = intformal( C/x * intformal( C +x*O(x^(2*n)) ) ); C = 1 + intformal( S/x * intformal( C +x*O(x^(2*n)) ) ); ); (2*n)!^2*polcoeff(C, 2*n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A325290 (C+S), A325292 (S). Cf. A326551. Sequence in context: A009555 A054959 A206305 * A326551 A253471 A230879 Adjacent sequences: A325288 A325289 A325290 * A325292 A325293 A325294 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 16 2019 STATUS approved

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Last modified July 22 16:15 EDT 2024. Contains 374540 sequences. (Running on oeis4.)