The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A097817 E.g.f. exp(2x)/(1-3x). 2
 1, 5, 34, 314, 3784, 56792, 1022320, 21468848, 515252608, 13911820928, 417354628864, 13772702754560, 495817299168256, 19336874667570176, 812148736037963776, 36546693121708402688, 1754241269842003394560 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Second binomial transform of n!3^n. LINKS FORMULA a(n) = 3n*a(n-1)+2^n, n>0, a(0)=1. a(n) ~ n! * exp(2/3) * 3^n. - Vaclav Kotesovec, Aug 04 2014 a(n) +(-3*n-2)*a(n-1) +6*(n-1)*a(n-2)=0. - R. J. Mathar, Dec 21 2014 From Peter Bala, Jan 30 2015: (Start) a(n) = int {x = 0..inf} (3*x + 2)^n*exp(-x) dx. The e.g.f. y = exp(2*x)/(1 - 3*x) satisfies the differential equation (1 - 3*x)*y' = (5 - 6*x)*y. Mathar's recurrence above follows easily from this. The sequence b(n) = 3^n*n! also satisfies Mathar's recurrence with b(0) = 1, b(1) = 3. This leads to the continued fraction representation a(n) = 3^n*n!*( 1 + 2/(3 - 6/(8 - 12/(11 - ... - (6*n - 6)/(3*n + 2) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(2/3) = 1 + 2/(3 - 6/(8 - 12/(11 - ... - (6*n - 6)/((3*n + 2) - ... )))). (End) a(n) = 3^n*exp(2/3)*Gamma(n+1,2/3). - Gerry Martens, Jul 24 2015 MATHEMATICA With[{nn=20}, CoefficientList[Series[Exp[2x]/(1-3x), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Apr 02 2020 *) CROSSREFS Cf. A010844, A097819. Sequence in context: A328488 A258179 A068475 * A303175 A197714 A198078 Adjacent sequences:  A097814 A097815 A097816 * A097818 A097819 A097820 KEYWORD easy,nonn AUTHOR Paul Barry, Aug 26 2004 STATUS approved

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

Last modified September 21 12:13 EDT 2020. Contains 337271 sequences. (Running on oeis4.)