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 A266490 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(2*x) * exp( Integral A(x) dx ), where the constant of integration is zero. 2
 1, 1, 4, 20, 126, 972, 8876, 93580, 1119328, 14986944, 222184136, 3614288272, 64022264176, 1226914925840, 25295189791296, 558317369479616, 13136590271813856, 328243850207690432, 8680766764223956416, 242245419192494844096, 7113910552105144027136, 219304957649505551899136, 7081169542830272102170752, 238996807468258679150596352 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Compare to: G(x) = exp( Integral G(x) dx ) when G(x) = 1/(1-x). LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..200 FORMULA E.g.f. A(x) satisfies: (1) A(x) = exp( Integral A(x) + 2*log(A(x)) dx ). (2) A(x) = A'(x)/A(x) - 2*log(A(x)). (3) log(A(x)) = exp(2*x) * Integral exp(-2*x)*A(x) dx. (4) A(x) = exp( Series_Reversion( Integral 1/(exp(x) + 2*x) dx ) ). a(n) ~ c^(n+1) * n!, where c = 1/Integral_{x=0..infinity} 1/(2*x + exp(x)) dx = 1.4650202775490107369040248583790383461628786237838809798971... - Vaclav Kotesovec, Aug 21 2017 EXAMPLE E.g.f.: A(x) = 1 + x + 4*x^2/2! + 20*x^3/3! + 126*x^4/4! + 972*x^5/5! + 8876*x^6/6! + 93580*x^7/7! + 1119328*x^8/8! + 14986944*x^9/9! + 222184136*x^10/10! +... such that log(A(x)) = Integral B(x) dx where B(x) = 1 + 3*x + 10*x^2/2! + 40*x^3/3! + 206*x^4/4! + 1384*x^5/5! + 11644*x^6/6! + 116868*x^7/7! + 1353064*x^8/8! + 17693072*x^9/9! + 257570280*x^10/10! +... and A(x) and B(x) satisfy: (1) A(x) = B'(x)/B(x) - 2, (2) B(x) = A'(x)/A(x), (3) B(x) = A(x) + 2*log(A(x)), (4) log(A(x)) = Integral B(x) dx, (5) log(B(x)) = Integral A(x) dx + 2*x. The Series Reversion of log(A(x)) equals Integral 1/(exp(x) + 2*x) dx: Integral 1/(exp(x) + 2*x) dx  =  x - 3*x^2/2! + 17*x^3/3! - 145*x^4/4! + 1649*x^5/5! - 23441*x^6/6! + 399865*x^7/7! - 7957881*x^8/8! + 180997857*x^9/9! - 4631289697*x^10/10! +... so that A( Integral 1/(exp(x) + 2*x) dx ) = exp(x). MATHEMATICA a[ n_] := a[n] = If[ n < 1, Boole[n == 0], Sum[ Binomial[n - 1, k - 1] a[n - k] Sum[ 2^(j - 1) a[k - j], {j, k}], {k, n}]]; (* Michael Somos, Aug 08 2017 *) 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( 2 + A ) ) ); 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)) + 2*x) ) )), n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A266328, A266329. Sequence in context: A297924 A151341 A285868 * A135886 A007550 A080795 Adjacent sequences:  A266487 A266488 A266489 * A266491 A266492 A266493 KEYWORD nonn AUTHOR Paul D. Hanna, Jan 27 2016 STATUS approved

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Last modified April 13 19:02 EDT 2021. Contains 342939 sequences. (Running on oeis4.)