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 A274276 E.g.f. A(x) satisfies: A( sqrt( A(x^2*exp(-2*x)) ) ) = x, where A(x) = Sum_{n>=1} a(n)*x^n/(n-1)!. 2
 1, 1, 2, 10, 80, 776, 8992, 130768, 2252672, 43823872, 957193856, 23369928704, 629680631296, 18514472015872, 590350181439488, 20311856724176896, 749913022501879808, 29561045244530032640, 1239353203580700000256, 55077035791625925492736, 2586090541400666789543936, 127922890235433583945056256, 6649362432158408977810522112, 362360171399316029979428126720, 20658795751396952768159379619840 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 1..200 (first 150 terms from Paul D. Hanna) FORMULA a(n) = A274275(n)/n. E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / (n-1)! satisfies: (1) A( sqrt( A(x^2*exp(2*x)) ) ) = -LambertW(-x*exp(x)). (2) A(x) = Series_Reversion( sqrt( A(x^2*exp(-x)) ) ). (3) A( A(x)^2 * exp(-2*A(x)) ) = x^2. (4) A(-A(x)^2 * exp(-2*A(x)) ) = -LambertW(x^2*exp(-x^2)). a(n)/n! ~ c * d^n / n^(5/2), where d = 2.52462188117..., c = 0.36965356... . - Vaclav Kotesovec, Jun 23 2016 EXAMPLE E.g.f.: A(x) = x + x^2 + 2*x^3/2! + 10*x^4/3! + 80*x^5/4! + 776*x^6/5! + 8992*x^7/6! + 130768*x^8/7! + 2252672*x^9/8! + 43823872*x^10/9! + 957193856*x^11/10! +... such that A( sqrt( A(x^2*exp(-2*x)) ) ) = x. PROG (PARI) {a(n) = my(A=x); for(i=1, n, A = serreverse( sqrt( subst(A, x, x^2*exp(-2*x +x*O(x^n))) ) ) ); (n-1)!*polcoeff(A, n)} for(n=1, 30, print1(a(n), ", ")) CROSSREFS Cf. A274275. Sequence in context: A108486 A152168 A003578 * A152600 A371460 A220112 Adjacent sequences: A274273 A274274 A274275 * A274277 A274278 A274279 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 17 2016 STATUS approved

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Last modified September 18 03:51 EDT 2024. Contains 375995 sequences. (Running on oeis4.)