The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A207214 E.g.f.: Sum_{n>=0} exp(n*x) * Product_{k=1..n} (exp(k*x) - 1). 2
 1, 1, 7, 85, 1759, 55621, 2501407, 151984645, 12004046719, 1196068161541, 146792747463007, 21762540250822405, 3834791755438306879, 792270319634586707461, 189687840256042278859807, 52103089179906338874671365, 16275196750916467736633834239 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Compare the e.g.f. to the identity: exp(-x) = Sum_{n>=0} exp(n*x) * Product_{k=1..n} (1 - exp(k*x)). LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..180 Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019. FORMULA E.g.f. A(x) satisfies: A(x) = exp(-x)*(2*G(x) - 1), where G(x) = Sum_{n>=0} Product_{k=1..n} (exp(k*x) - 1) = e.g.f. of A158690. a(n) ~ sqrt(2) * 12^(n+1) * (n!)^2 / Pi^(2*n+2). - Vaclav Kotesovec, May 05 2014 EXAMPLE E.g.f.: A(x) = 1 + x + 7*x^2/2! + 85*x^3/3! + 1759*x^4/4! + 55621*x^5/5! +... such that, by definition, A(x) = 1 + exp(x) * (exp(x)-1) + exp(2*x) * (exp(x)-1)*(exp(2*x)-1) + exp(3*x) * (exp(x)-1)*(exp(2*x)-1)*(exp(3*x)-1) + exp(4*x) * (exp(x)-1)*(exp(2*x)-1)*(exp(3*x)-1)*(exp(4*x)-1) +... The related e.g.f. of A158690 equals the series: G(x) = 1 + (exp(x)-1) + (exp(x)-1)*(exp(2*x)-1) + (exp(x)-1)*(exp(2*x)-1)*(exp(3*x)-1) + (exp(x)-1)*(exp(2*x)-1)*(exp(3*x)-1)*(exp(4*x)-1) +... or, more explicitly, G(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1073*x^4/4! + 32671*x^5/5! +... such that G(x) satisfies: G(x) = (1 + exp(x)*A(x))/2. PROG (PARI) {a(n)=n!*polcoeff(sum(m=0, n+1, exp(m*x+x*O(x^n))*prod(k=1, m, exp(k*x+x*O(x^n))-1)), n)} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A158690. Sequence in context: A121020 A060237 A000424 * A000686 A102923 A220246 Adjacent sequences: A207211 A207212 A207213 * A207215 A207216 A207217 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 16 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 21 07:34 EDT 2023. Contains 361393 sequences. (Running on oeis4.)