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 A266328 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(-x) * exp( Integral A(x) dx ), where the constant of integration is zero. 4
 1, 1, 1, 2, 6, 21, 92, 469, 2731, 17985, 131528, 1059616, 9319363, 88833422, 912393381, 10043727089, 117969438513, 1472593659884, 19467505081458, 271704942613323, 3992343851680466, 61603531051030691, 995949139457447931, 16835191741257445589, 296976010796327785530, 5457427389713208932740, 104308245862443706265341, 2070461793105333579698992, 42622090166454492404075635 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Compare to: G(x) = exp( Integral G(x) dx ) when G(x) = 1/(1-x). What is lim_{n->oo} (a(n)/n!)^(1/n)? Example: (a(500)/500!)^(1/500) = 0.7353325805... Limit_{n->oo} (a(n)/n!)^(1/n) = 1/Integral_{x=0..oo} 1/(exp(x) - x) dx = 0.73578196429164719984313538... - Vaclav Kotesovec, Aug 21 2017 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..300 FORMULA E.g.f. A(x) satisfies: (1) A(x) = exp( Integral A(x) - log(A(x)) dx ). (2) A(x) = log(A(x)) + A'(x)/A(x). (3) log(A(x)) = exp(-x) * Integral exp(x)*A(x) dx. (4) A(x) = exp( Series_Reversion( Integral 1/(exp(x) - x) dx ) ). a(n) ~ c^(n+1) * n!, where c = 1/Integral_{x=0..oo} 1/(exp(x) - x) dx = 0.7357819642916471998431353808137704665788888148929882090175... - Vaclav Kotesovec, Aug 21 2017 Conjecture: a(n) = R(n-1, 0) for n > 0 with a(0) = 1 where R(n, q) = R(n-1, q+1) + Sum_{j=0..q-1} binomial(q+1, j)*R(n-1, j) for n > 0, q >= 0 with R(0, q) = 1 for q >= 0. - Mikhail Kurkov, Dec 26 2023 EXAMPLE E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 6*x^4/4! + 21*x^5/5! + 92*x^6/6! + 469*x^7/7! + 2731*x^8/8! + 17985*x^9/9! + 131528*x^10/10! + ... such that log(A(x)) = Integral B(x) dx where B(x) = 1 + x^2/2! + x^3/3! + 5*x^4/4! + 16*x^5/5! + 76*x^6/6! + 393*x^7/7! + 2338*x^8/8! + 15647*x^9/9! + 115881*x^10/10! + ... and A(x) and B(x) satisfy: (1) A(x) = B'(x)/B(x) + 1, (2) B(x) = A'(x)/A(x), (3) B(x) = A(x) - log(A(x)), (4) log(A(x)) = Integral B(x) dx, (5) log(B(x)) = Integral A(x) dx - x. The Series Reversion of log(A(x)) equals Integral 1/(exp(x) - x) dx: Integral 1/(exp(x) - x) dx = x - x^3/3! - x^4/4! + 5*x^5/5! + 19*x^6/6! - 41*x^7/7! - 519*x^8/8! - 183*x^9/9! + 19223*x^10/10! + ... + A089148(n-1)*x^n/n! + ... so that A( Integral 1/(exp(x) - x) dx ) = exp(x). PROG (PARI) {a(n) = my(A=1+x, B=1+x); for(i=0, n, A = exp( intformal( B + x*O(x^n) ) ); B = exp( intformal( A - 1 ) ) ); n!*polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) (PARI) {a(n) = n! * polcoeff( exp( serreverse( intformal( 1/(exp(x +x*O(x^n)) - x) ) )), n)} for(n=0, 30, print1(a(n), ", ")) (PARI) Vec( serlaplace( exp( serreverse( intformal( 1/(exp(x +x*O(x^25)) - x)))))) \\ Joerg Arndt, Dec 26 2023 CROSSREFS Cf. A006789, A089148, A266329. Sequence in context: A231166 A115089 A304196 * A214087 A183950 A001928 Adjacent sequences: A266325 A266326 A266327 * A266329 A266330 A266331 KEYWORD nonn AUTHOR Paul D. Hanna, Jan 24 2016 STATUS approved

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Last modified September 12 12:03 EDT 2024. Contains 375851 sequences. (Running on oeis4.)