This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A245313 G.f. satisfies: A(x) = 1 + x*A(x) + x^2*A(x)^2 + 3*x^3*A(x)*A'(x) + x^4*A(x)*A''(x). 3
 1, 1, 2, 7, 31, 176, 1158, 8919, 76751, 742597, 7865088, 91553100, 1150905332, 15665172108, 227991734414, 3554320236911, 58795765799791, 1033303679424539, 19151079894682674, 374662948814998855, 7691131223011551255, 165785969673935575904, 3734170668741419488552 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Do the following limits exist? If so, what are the respective values? (1) limit a(n)*sqrt(n+1)/(n+1)! ? (Value is near 0.718490 at n=400.) (2) limit A245313(n)/A245312(n) ? (Value is near 2.721747 at n=400.) Limit a(n)*sqrt(n+1)/(n+1)! = 0.7189460513696389360211370..., limit A245313(n)/A245312(n) = e. - Vaclav Kotesovec, Jul 20 2014 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..400 FORMULA G.f. A(x) satisfies: (1) A(x) = 1/(1-x - x^2*Dx^2(A(x))) where Dx(F(x)) = d/dx x*F(x). (2) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A245312. (3) A(x) = (1/x)*Series_Reversion(x/G(x)) where G(x) is the g.f. of A245312. a(n) = [x^n] G(x)^(n+1)/(n+1) for n>=0 where G(x) is the g.f. of A245312. a(n) = [x^(n+2)] G(x)^(n+1)/(n+1)^3 for n>=0 where G(x) is the g.f. of A245312. EXAMPLE G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 176*x^5 + 1158*x^6 +... where A(x) = 1 / (1-x - x^2*A(x) - 3*x^3*A'(x) - x^4*A''(x)). Define Dx(A(x)) = d/dx x*A(x) = A(x) + x*A'(x): Dx(A(x)) = 1 + 2*x + 6*x^2 + 28*x^3 + 155*x^4 + 1056*x^5 + 8106*x^6 +... so that Dx^2(A(x)) = d/dx x*(d/dx x*A(x)) = A(x) + 3*x*A'(x) + x^2*A''(x): Dx^2(A(x)) = 1 + 4*x + 18*x^2 + 112*x^3 + 775*x^4 + 6336*x^5 +... then A(x) = 1/(1-x - x^2*Dx^2(A(x))): 1/A(x) = 1 - x - x^2 - 4*x^3 - 18*x^4 - 112*x^5 - 775*x^6 - 6336*x^7 -... RELATED SERIES. The g.f. of A245312 begins: G(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 60*x^5 + 360*x^6 + 2940*x^7 +... where A(x) = G(x*A(x)) where G(x) = A(x/G(x)) and a(n) = [x^n] G(x)^(n+1)/(n+1) = [x^(n+2)] G(x)^(n+1)/(n+1)^3 for n>=0. PROG (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A+x^2*A^2+3*x^3*A*A'+x^4*A*A''+x*O(x^n)); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) (PARI) /* From A(x) = G(x*A(x)) where G(x) is the g.f. of A245312: */ {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); m=#A-2; A[#A]=-Vec(Ser(A)^m*(1-m^2*x^2))[#A]/m); polcoeff(1/x*serreverse(x/Ser(A)), n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A245312, A245311. Sequence in context: A005977 A199675 A059037 * A306037 A046907 A091312 Adjacent sequences:  A245310 A245311 A245312 * A245314 A245315 A245316 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 19 2014 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 December 15 09:05 EST 2019. Contains 329995 sequences. (Running on oeis4.)